Tzg3DuNz

1{
2  "cells": [
3    {
4      "metadata": {},
5      "cell_type": "markdown",
6      "source": "# Limits\nYou can use algebraeic methods to calculate the rate of change over a function interval by joining two points on the function with a secant line and measuring its slope. For example, a function might return the distance travelled by a cyclist in a period of time, and you can use a secant line to measure the average velocity between two points in time. However, this doesn't tell you the cyclist's vecolcity at any single point in time - just the average speed over an interval.\n\nTo find the cyclist's velocity at a specific point in time, you need the ability to find the slope of a curve at a given point. *Differential Calculus* enables us to do through the use of *derivatives*. We can use derivatives to find the slope at a specific *x* value by calculating a delta for *x<sub>1</sub>* and *x<sub>2</sub>* values that are infinitesimally close together - so you can think of it as measuring the slope of a tiny straight line that comprises part of the curve.\n\n## Introduction to Limits\nHowever, before we can jump straight into derivatives, we need to examine another aspect of differential calculus - the *limit* of a function; which helps us measure how a function's value changes as the *x<sub>2</sub>* value approaches *x<sub>1</sub>*\n\nTo better understand limits, let's take a closer look at our function, and note that although we graph the function as a line, it is in fact made up of individual points. Run the following cell to show the points that we've plotted for integer values of ***x*** - the line is created by interpolating the points in between:"
7    },
8    {
9      "metadata": {
10        "trusted": true
11      },
12      "cell_type": "code",
13      "source": "%matplotlib inline\n\n# Here's the function\ndef f(x):\n    return x**2 + x\n\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values from 0 to 10 to plot\nx = list(range(0, 11))\n\n# Get the corresponding y values from the function\ny = [f(i) for i in x] \n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('f(x)')\nplt.grid()\n\n# Plot the function\nplt.plot(x,y, color='lightgrey', marker='o', markeredgecolor='green', markerfacecolor='green')\n\nplt.show()",
14      "execution_count": 1,
15      "outputs": [
16        {
17          "output_type": "display_data",
18          "data": {
19            "image/png": 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\n",
20            "text/plain": "<matplotlib.figure.Figure at 0x7fb64eae0ba8>"
21          },
22          "metadata": {}
23        }
24      ]
25    },
26    {
27      "metadata": {},
28      "cell_type": "markdown",
29      "source": "We know from the function that the ***f(x)*** values are calculated by squaring the ***x*** value and adding ***x***, so we can easily calculate points in between and show them - run the following code to see this:"
30    },
31    {
32      "metadata": {
33        "trusted": true
34      },
35      "cell_type": "code",
36      "source": "%matplotlib inline\n\n# Here's the function\ndef f(x):\n    return x**2 + x\n\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values from 0 to 10 to plot\nx = list(range(0,5))\nx.append(4.25)\nx.append(4.5)\nx.append(4.75)\nx.append(5)\nx.append(5.25)\nx.append(5.5)\nx.append(5.75)\nx = x + list(range(6,11))\n\n# Get the corresponding y values from the function\ny = [f(i) for i in x] \n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('f(x)')\nplt.grid()\n\n# Plot the function\nplt.plot(x,y, color='lightgrey', marker='o', markeredgecolor='green', markerfacecolor='green')\n\nplt.show()",
37      "execution_count": 2,
38      "outputs": [
39        {
40          "output_type": "display_data",
41          "data": {
42            "image/png": 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e/g/g9e7+eC4LE5HC5ZnZ0Xe9/C7qKuvOey5UHWLvbr3FVApW+hbT7cDngd8G/hr4mpm9Mod1iUiBWlhYYHh4mLGxMW575W0M3DxAJBzBMCLhCAM3D7Dn+j1BlynrYKVLKf4X4Jfc/STwiJn9A4tB8apcFSYihWdubo5YLMbMzAwdHR00NzdzW+dt3PbK24IuTXJgRQHh7rde8PjfzewXclOSiBSis2fPEovFcHcikQiNjY1BlyQ5dtm3mMzsj80s60wXd58zs11mdlNuShORQnHmzJlze0e/7GUvUziUiSvdQRwCHjezGeC7LC7TXQtsB/4z8E3gIzmtUEQC4+4cP36c06dPU19fT09Pj/ZxKCNXCohfd/cbzeyPWFxmo4PFpbn/Fuhz9+lcFygiwVjqjE4kErS0tGjBvTJ0pYC4wcwiwB5g5wXP1bG4cJ+IlJjZ2VlisRizs7NaU6mMXSkg/gr4Oou7wD277LgBnjkuIiUkkUgwPDwMwLZt26ivrw+4IgnKZQPC3R8CHjKzv3T3389TTSISAHdnfHycY8eOsWHDBiKRCDU1NUGXJQFa6TBXhYNICXN3RkdHOXPmDI2NjXR1dakzWlY8UU5ESlQqlSIWi5FMJtm8eTNtbW3qjBZAASFS1mZmZhgaGiKVStHV1UVTU1PQJUkBUUCIlKnJyUlGRkaoqKhg27ZthEKhoEuSAqOAECkz7s6pU6c4ceIEtbW1RCIRqqurgy5LCpACQqSMpNNpRkdHmZiYIBwO09nZSUXFSreFkXKjgBApE/Pz88RiMaanp2ltbWXLli3qjJbLUkCIlIHp6WmGhoZYWFigu7ubcDh85RdJ2VNAiJS4eDzOyMgIVVVVXHPNNdTV1V35RSIoIERKlrtz8uRJxsbGCIVC9PT0UFWl/+Vl5dQ7JVIiBg8NEt0XZddTu4jsi/DnB/+csbExmpqaiEajCgdZNQWESAkYPDRI3+N9DMWHcJxYPMY937qHpyef1kglWTP91IiUgP4D/STnk+cdm1mY4cF/f1AjlWTNAgsIM6s0s++Z2Vczj7eZ2TNm9qKZ/b2ZaRlJkRWKxWOrOi6yEkHeQbwXeH7Z448Cf+bu24EzwDsCqUqkyCwsLLC1fmvW53rCPXmuRkpJIAFhZl3AW4DPZh4bsAt4NHPKw8CtQdQmUkySySSHDx/m7mvvpq7q/OGroeoQe3fvDagyKQVB3UHsA/4ISGcetwAT7p7KPB4BOoMoTKRQLY1SqvhwBZF9ET79r5/myJEjANy9424+89bPEAlHMIxIOMLAzQPsuX5PwFVLMTN3z+8FzW4C3uzuf2BmO4D/BtwJ/Ju7/1TmnG7ga+5+fZbX9wF9AG1tbTfs379/TXUkEgkaGhrW1ogipTYXr2+e+CYf+38fYzY9e+5YbWUt/df387qm17H8/+NSafNqqM2rs3PnzufcvfdK5wUREA8AtwMpoBbYCPwD8Eag3d1TZvaLwH939zde7mv19vb6s88+e7lTLungwYPs2LFjTa8tVmpz8YruizIUH7roeE+4h6H3nX+8VNq8Gmrz6pjZigIi728xufuH3L3L3aPAbwBPuPse4Eng1zOn3QF8Jd+1iRSqS41GGo4P57kSKSeFNA/iHuADZnaYxT6JzwVcj0hBmJ+fp6O+I+tzGqUkuRRoQLj7QXe/KfP5T9z91e7+U+7+dnefvdLrRUrd1NQUhw8f5j3/6T0apSR5V0h3ECJlb/lIpa4/7eJT//Ipqquree/O92qUkuSdVu8SKRBL6yktLZnxUuIlPvy9D9PZ2clttbex5/o9CgTJK91BiBSI+w7cd9F6StOpaf74yT8OqCIpdwoIkQIwPz9/yRFJWk9JgqKAEAlYPB7n8OHDtIfasz6vkUoSFAWESB5duFzGQwcfYnh4mJqaGh7Y/QCh6tB552ukkgRJASGSJ1k39Xn6Hv5l4l+45ppruOPn7mDg5gGNVJKCoVFMInlyqU19PvrsR/n9X/p9AI1UkoKiOwiRPNGmPlJsFBAiOZZOpxkdHaW9Tp3QUlwUECLraHkndHRflM9/5/O8+OKLjI+P0/+afnVCS1FRQIiskws7oYfiQ9z1j3fx1aGvsm3bNu56/V3qhJaiok5qkXVyqU7oh378EB/81Q8C6oSW4qI7CJF1oj0bpNQoIESukrtz5swZzYSWkqOAEFml82ZD/1mETzzxCV566SX+8FV/SKhKndBSOhQQIqtw0WzoyRgf+rcP8a3Jb/G+3e9j4K3qhJbSoU5qkVW4VEf0A//+AL/72t9VJ7SUFN1BiKxQKpXSbGgpKwoIkQtcONlt8AeDjI+P8+KLL2o2tJQVBYTIMtkmu73zsXfymW9/htraWj6y+yOaDS1lQwEhssyl+hj+4oW/IBqN8js3/I5mQ0vZUCe1yDKX6kt4aeolzAzQbGgpH7qDEMlIJBJ0hDqyPqc+BilHCggpS9kmux09epT3/+z7qauqO+9c9TFIuVJASNnJNtnt3n+9l6fjT/OBN3yAz7z1M+pjEEF9EFKG7jtwX9aO6Ae/8yC/d+PvqY9BJCPvdxBm1m1mT5rZ82b2IzN7b+Z4s5l9w8xezHzclO/apLS5O5OTk5dcXVWT3UTOF8RbTCngg+7+CuA1wF1mdi1wL3DA3bcDBzKPRdZkqY9h11O7iO6L8oVnv8CRI0eIxWLqiBZZobwHhLsfc/fvZj6fAp4HOoFbgIczpz0M3Jrv2qQ0ZJvs9gdf/wO+9OKX2Lp1K3/yxj/RZDeRFQi0k9rMosCrgGeANnc/BoshArQGV5kUs0tOdnv+L2hubmbPz+7RZDeRFTB3D+bCZg3AU8Bed/+ymU24e9Oy58+4+0X9EGbWB/QBtLW13bB///41XT+RSNDQ0LC24otUObS5srKSX37il3Eu/rk2jCd++YkAqsqvcvg+X0htXp2dO3c+5+69VzovkFFMZlYNfAkYdPcvZw6fMLMOdz9mZh3AyWyvdfcBYACgt7fXd+zYsaYaDh48yFpfW6xKtc3uTjKZZGxsjEQiQXuonWPJYxed1xPuKcn2X6hUv8+XozbnRhCjmAz4HPC8u3982VOPAXdkPr8D+Eq+a5PisjQq6ciRIxw5coTp6WlaW1v56K9+VH0MIusgiDuIG4HbgUNm9v3MsfuAB4Evmtk7gBjw9gBqkyLg7sTjccbGxpidnaW6upqOjg42bdpERUUFt7feTkVFBf0H+onFY/SEe9i7e6/6GERWKe8B4e5PA3aJp3fnsxYpLul0momJCcbGxpifn2fDhg10dnbS1NR0biG9JUuT3crxrQeR9aKZ1FLwFhYWGB8f5/Tp06RSKerq6ujo6KCxsfGiYBCR9aOAkIKVSqU4ffo0p0+fJp1OU19fT1dXF/X19QoGkTxQQEjBmZub49SpU5w5cwZ3Z+PGjWzZsoW6urorv1hE1o0CQgrGzMwMp06dYmJiAoCmpia2bNnChg0bAq5MpDwpICRwS3MYpqamMDNaWlpoaWmhpqYm6NJEypoCQgLh7pw9e5axsTHOnj1LZWUlW7ZsoaWlhaoq/ViKFAL9nyg5NXho8Pz5CLv2clPkJsbGxpiZmaGqqor29nY2bdpEZWVl0OWKyDLaUU5yJtuqqu987J381bf+inQ6zdatW/npn/5pNm/erHAQKUAKCMmZS62q+skXPsn27dtpbm6mokI/giKFSm8xybpLp9NMTk5ecoe2kakRzWMQKQIKCFkX7k4ikWBiYoKpqSnS6TQdoQ5Gk6MXnaud20SKg+7vZc2WltkeHR3lhRdeYGhoiEQiQTgcZtu2bdq5TaTI6Q5CVm1mZoZ4PM7ExATz8/OYGY2NjTQ1NdHQ0HCuX2HPz+4BQ6uqihQpBYSsyPz8/LlQmJmZAaChoYHW1lY2btx4yVFIS6uqikjxUUDIJS0sLBCPx4nH45w9exaAuro62tvbCYfDVFdXB1yhiOSSAkLOk06nmZqaIh6PMzU1hbtTU1NDa2sr4XBY6yKJlBEFhJxb9mJiYoLJyUnS6TRVVVU0NzfT1NREbW2thqWKlCEFRBk4b7mL7y92FP/Wdb/FzMwMExMTxONxUqkUFRUVbNy4kaamJu25ICIKiFK3tNzF0ozmofgQ73rsXRw/dpw3bn0jZkZDQwNNTU00NjZqZrOInKOAKHH3HbjvouUuplPTfPwHH+fO3jsJh8NaB0lEslJAlJhUKkUymSSZTHL27FmG48NZzzt29hjNzc15rk5EiokCosjNzc2dC4NkMsns7CwAZkZdXR2dDZ2MJEYuep2WuxCRK1FAFBF3Z3Z29lwYJJNJ5ufnAaioqCAUChEOh6mvr6euro6Kigoe/NUHz+uDAC13ISIro4AoYOl0munp6fPeMkqn0wBUVVVRX19PKBQiFApdcijq0ixmLXchIqulgMiji3ZXu+AX9cLCwnlvF01PT+PuAGzYsIFwOEwoFKK+vp7q6uoVD0NdWu7i4MGD7NixIxdNE5ESpIDIk2zDTfse6yN5Nslbet5CMpk8t8YRLC5p0dzcfO4uQfs0i0i+6bdOHiwsLPChb37oouGmyVSS+//5fm686Ubq6upobW0995aR5iOISNAKKiDM7E3AJ4BK4LPu/mDAJV2Ru5NKpZifn2dubo75+flz/5Yep9NpRiYvHkkEcHz6OK94xSs0a1lECk7BBISZVQKfBN4AjADfMbPH3P3H63mdbMtOXK7DdmFh4bxf+tmC4EKVlZVUV1dTU1Nzrr+gq7GL4amL5yT0hHsUDiJSkAomIIBXA4fd/ScAZrYfuAVYt4C4VD/A3Owcb9v+tot+8c/Pz7OwsHDR16murqa6uppQKERNTc25x0v/ss1MfuAND2i4qYgUlUIKiE5g+Z/YI8AvrOcF+g/0Z+0H6H+in5+v/XlgcT7B0l//oVDovF/8NTU1VFVVrekvfg03FZFiY0vDKINmZm8H3uju78w8vh14tbvffcF5fUAfQFtb2w379+9f8TV2PbUL5+L2GsZTu54inU5TKP89ciGRSNDQ0BB0GXmlNpcHtXl1du7c+Zy7917pvEK6gxgBupc97gJGLzzJ3QeAAYDe3l5fzbj+nu/3MBQfuvh4uIfXve51qyy3+JTjPAi1uTyozblRSGMpvwNsN7NtZlYD/Abw2HpeYO/uvYSqQ+cdUz+AiEh2BRMQ7p4C3g38I/A88EV3/9F6XmPP9XsYuHmASDiCYUTCEQZuHlA/gIhIFoX0FhPu/jXga7m8hpadEBFZmYK5gxARkcKigBARkawUECIikpUCQkREslJAiIhIVgUzk3otzGwMuHjm28psBk6tYznFQG0uD2pzebiaNkfcfcuVTirqgLgaZvbsSqaalxK1uTyozeUhH23WW0wiIpKVAkJERLIq54AYCLqAAKjN5UFtLg85b3PZ9kGIiMjllfMdhIiIXEZZBoSZvcnM/sPMDpvZvUHXk2tm1m1mT5rZ82b2IzN7b9A15YOZVZrZ98zsq0HXkg9m1mRmj5rZC5nv9S8GXVOumdn7Mz/TPzSzR8ysNuia1puZfd7MTprZD5cdazazb5jZi5mPm3Jx7bILCDOrBD4J/BpwLfCbZnZtsFXlXAr4oLu/AngNcFcZtBngvSwuHV8uPgF83d1fDrySEm+7mXUC7wF63f06oJLFfWRKzV8Db7rg2L3AAXffDhzIPF53ZRcQwKuBw+7+E3efA/YDtwRcU065+zF3/27m8ykWf3F0BltVbplZF/AW4LNB15IPZrYReD3wOQB3n3P3iWCryosqoM7MqoAQWXahLHbu/s/A+AWHbwEeznz+MHBrLq5djgHRCQwvezxCif+yXM7MosCrgGeCrSTn9gF/BKSDLiRPrgHGgC9k3lb7rJnVB11ULrn7S8DHgBhwDIi7+z8FW1XetLn7MVj8AxBozcVFyjEgLMuxshjKZWYNwJeA97n7ZND15IqZ3QScdPfngq4lj6qAnwP+0t1fBZwlR287FIrM++63ANuArUC9md0WbFWlpRwDYgToXva4ixK8Lb2QmVWzGA6D7v7loOvJsRuBt5rZURbfQtxlZn8bbEk5NwKMuPvSneGjLAZGKfsV4Ii7j7n7PPBl4LUB15QvJ8ysAyDz8WQuLlKOAfEdYLtCf+hOAAABpUlEQVSZbTOzGhY7tR4LuKacMjNj8b3p593940HXk2vu/iF373L3KIvf3yfcvaT/snT348Cwmf1M5tBu4McBlpQPMeA1ZhbK/IzvpsQ75pd5DLgj8/kdwFdycZGC2pM6H9w9ZWbvBv6RxVEPn3f3HwVcVq7dCNwOHDKz72eO3ZfZA1xKx93AYOYPn58AdwZcT065+zNm9ijwXRZH6n2PEpxRbWaPADuAzWY2AtwPPAh80czewWJQvj0n19ZMahERyaYc32ISEZEVUECIiEhWCggREclKASEiIlkpIEREJCsFhIiIZKWAEBGRrBQQIuvIzH7ezH5gZrVmVp/Zq+C6oOsSWQtNlBNZZ2b2P4FaoI7F9ZEeCLgkkTVRQIiss8xSF98BZoDXuvtCwCWJrIneYhJZf81AA9DI4p2ESFHSHYTIOjOzx1hcZnwb0OHu7w64JJE1KbvVXEVyycx+G0i5+99l9j//VzPb5e5PBF2byGrpDkJERLJSH4SIiGSlgBARkawUECIikpUCQkREslJAiIhIVgoIERHJSgEhIiJZKSBERCSr/w8J+qFUlDy8PQAAAABJRU5ErkJggg==\n",
43            "text/plain": "<matplotlib.figure.Figure at 0x7fb64eae0c50>"
44          },
45          "metadata": {}
46        }
47      ]
48    },
49    {
50      "metadata": {},
51      "cell_type": "markdown",
52      "source": "Now we can see more clearly that this function line is formed of a continuous series of points, so theoretically for any given value of ***x*** there is a point on the line, and there is an adjacent point on either side with a value that is as close to ***x*** as possible, but not actually ***x***.\n\nRun the following code to visualize a specific point for *x = 5*, and try to identify the closest point either side of it:"
53    },
54    {
55      "metadata": {
56        "trusted": true
57      },
58      "cell_type": "code",
59      "source": "%matplotlib inline\n\n# Here's the function\ndef f(x):\n    return x**2 + x\n\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values from 0 to 10 to plot\nx = list(range(0,5))\nx.append(4.25)\nx.append(4.5)\nx.append(4.75)\nx.append(5)\nx.append(5.25)\nx.append(5.5)\nx.append(5.75)\nx = x + list(range(6,11))\n\n# Get the corresponding y values from the function\ny = [f(i) for i in x] \n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('f(x)')\nplt.grid()\n\n# Plot the function\nplt.plot(x,y, color='lightgrey', marker='o', markeredgecolor='green', markerfacecolor='green')\n\nzx = 5\nzy = f(zx)\nplt.plot(zx, zy, color='red', marker='o', markersize=10)\nplt.annotate('x=' + str(zx),(zx, zy), xytext=(zx - 0.5, zy + 5))\n\n# Plot f(x) when x = 5.1\nposx = 5.25\nposy = f(posx)\nplt.plot(posx, posy, color='blue', marker='<', markersize=10)\nplt.annotate('x=' + str(posx),(posx, posy), xytext=(posx + 0.5, posy - 1))\n\n# Plot f(x) when x = 4.9\nnegx = 4.75\nnegy = f(negx)\nplt.plot(negx, negy, color='orange', marker='>', markersize=10)\nplt.annotate('x=' + str(negx),(negx, negy), xytext=(negx - 1.5, negy - 1))\n\nplt.show()",
60      "execution_count": 3,
61      "outputs": [
62        {
63          "output_type": "display_data",
64          "data": {
65            "image/png": 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\n",
66            "text/plain": "<matplotlib.figure.Figure at 0x7fb64cbea3c8>"
67          },
68          "metadata": {}
69        }
70      ]
71    },
72    {
73      "metadata": {},
74      "cell_type": "markdown",
75      "source": "You can see the point where ***x*** is 5, and you can see that there are points shown on the graph that appear to be right next to this point (at *x=4.75* and *x=5.25*). However, if we zoomed in we'd see that there are still gaps that could be filled by other values of ***x*** that are even closer to 5; for example, 4.9 and 5.1, or 4.999 and 5.001. If we could zoom infinitely close to the line we'd see that no matter how close a value you use (for example, 4.999999999999), there is always a value that's fractionally closer (for example, 4.9999999999999).\n\nSo what we can say is that there is a hypothetical number that's as close as possible to our desired value of *x* without actually being *x*, but we can't express it as a real number. Instead, we express its symbolically as a *limit*, like this:\n\n\\begin{equation}\\lim_{x \\to 5} f(x)\\end{equation}\n\nThis is interpreted as *the limit of function f(x) as *x* approaches 5*."
76    },
77    {
78      "metadata": {},
79      "cell_type": "markdown",
80      "source": "##  Limits and Continuity\nThe function ***f(x)*** is *continuous* for all real numbered values of ***x***. Put simply, this means that you can draw the line created by the function without lifting your pen (we'll look at a more formal definition later in this course).\n\nHowever, this isn't necessarily true of all functions. Consider function ***g(x)*** below: \n\n\\begin{equation}g(x) = -(\\frac{12}{2x})^{2}\\end{equation}\n\nThis function is a little more complex than the previous one, but the key thing to note is that it requires a division by *2x*. Now, ask yourself; what would happen if you applied this function to an *x* value of **0**?\n\nWell, 2 &bull; 2 is 0, and anything divided by 0 is *undefined*. So the *domain* of this function does not include 0; in other words, the function is defined when *x* is any real number such that *x is not equal to 0*. The function should therefore be written like this:\n\n\\begin{equation}g(x) = -(\\frac{12}{2x})^{2},\\;\\; x \\ne 0\\end{equation}\n\nSo why is this important? Let's investigate by running the following Python code to define the function and plot it for a set of arbitrary of values:"
81    },
82    {
83      "metadata": {
84        "scrolled": false,
85        "trusted": true
86      },
87      "cell_type": "code",
88      "source": "%matplotlib inline\n\n# Define function g\ndef g(x):\n    if x != 0:\n        return -(12/(2*x))**2\n    \n# Plot output from function g\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-20, 21)\n\n# Get the corresponding y values from the function\ny = [g(a) for a in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('g(x)')\nplt.grid()\n\n# Plot x against g(x)\nplt.plot(x,y, color='green')\n\nplt.show()",
89      "execution_count": 4,
90      "outputs": [
91        {
92          "output_type": "display_data",
93          "data": {
94            "image/png": 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\n",
95            "text/plain": "<matplotlib.figure.Figure at 0x7fb64d240470>"
96          },
97          "metadata": {}
98        }
99      ]
100    },
101    {
102      "metadata": {},
103      "cell_type": "markdown",
104      "source": "Look closely at the plot, and note the gap the line where *x* = 0. This indicates that the function is not defined here.The *domain* of the function (it's set of possible input values) not include 0, and it's *range* (the set of possible output values) does not include a value for x=0.\n\nThis is a *non-continuous* function - in other words, it includes at least one gap when plotted (so you couldn't plot it by hand without lifting your pen). Specifically, the function is non-continuous at x=0.\n\nBy convention, when a non-continuous function is plotted, the points that form a continuous line (or *interval*) are shown as a line, and the end of each line where there is a discontinuity is shown as a circle, which is filled if the value at that point is included in the line and empty if the value is not included in the line.\n\nIn this case, the function produces two intervals with a gap between them where the function is not defined, so we can show the discontinuous point as an unfilled circle - run the following code to visualize this with Python:"
105    },
106    {
107      "metadata": {
108        "scrolled": false,
109        "trusted": true
110      },
111      "cell_type": "code",
112      "source": "%matplotlib inline\n\n# Define function g\ndef g(x):\n    if x != 0:\n        return -(12/(2*x))**2\n    \n# Plot output from function g\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-20, 21)\n\n\n# Get the corresponding y values from the function\ny = [g(a) for a in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('g(x)')\nplt.grid()\n\n# Plot x against g(x)\nplt.plot(x,y, color='green')\n\n# plot a circle at the gap (or close enough anyway!)\nxy = (0,g(1))\nplt.annotate('O',xy, xytext=(-0.7, -37),fontsize=14,color='green')\n\nplt.show()",
113      "execution_count": 5,
114      "outputs": [
115        {
116          "output_type": "display_data",
117          "data": {
118            "image/png": 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\n",
119            "text/plain": "<matplotlib.figure.Figure at 0x7fb64d240b00>"
120          },
121          "metadata": {}
122        }
123      ]
124    },
125    {
126      "metadata": {},
127      "cell_type": "markdown",
128      "source": "There are a number of reasons a function might be non-continuous. For example, consider the following function:\n\n\\begin{equation}h(x) = 2\\sqrt{x},\\;\\; x \\ge 0\\end{equation}\n\nApplying this function to a non-negative ***x*** value returns a valid output; but for any value where ***x*** is negative, the output is undefined, because the square root of a negative value is not a real number.\n\nHere's the Python to plot function ***h***:"
129    },
130    {
131      "metadata": {
132        "trusted": true
133      },
134      "cell_type": "code",
135      "source": "%matplotlib inline\n\ndef h(x):\n    if x >= 0:\n        import numpy as np\n        return 2 * np.sqrt(x)\n\n# Plot output from function h\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-20, 21)\n\n# Get the corresponding y values from the function\ny = [h(a) for a in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('h(x)')\nplt.grid()\n\n# Plot x against h(x)\nplt.plot(x,y, color='green')\n\n# plot a circle close enough to the h(-x) limit for our purposes!\nplt.plot(0, h(0), color='green', marker='o', markerfacecolor='green', markersize=10)\n\nplt.show()",
136      "execution_count": 6,
137      "outputs": [
138        {
139          "output_type": "display_data",
140          "data": {
141            "image/png": 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\n",
142            "text/plain": "<matplotlib.figure.Figure at 0x7fb64ca564a8>"
143          },
144          "metadata": {}
145        }
146      ]
147    },
148    {
149      "metadata": {},
150      "cell_type": "markdown",
151      "source": "Now, suppose we have a function like this:\n\n\\begin{equation}\nk(x) = \\begin{cases}\n  x + 20, & \\text{if } x \\le 0, \\\\\n  x - 100, & \\text{otherwise }\n\\end{cases}\n\\end{equation}\n\nIn this case, the function's domain includes all real numbers, but its output is still non-continuous because of the way different values are returned depending on the value of *x*. The *range* of possible outputs for *k(x &le; 0)* is &le; 20, and the range of output values for *k(x > 0)* is x > -100.\n\nLet's use Python to plot function ***k***:"
152    },
153    {
154      "metadata": {
155        "trusted": true
156      },
157      "cell_type": "code",
158      "source": "%matplotlib inline\n\ndef k(x):\n    import numpy as np\n    if x <= 0:\n        return x + 20\n    else:\n        return x - 100\n\n# Plot output from function h\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values for each non-contonuous interval\nx1 = range(-20, 1)\nx2 = range(1, 20)\n\n# Get the corresponding y values from the function\ny1 = [k(i) for i in x1]\ny2 = [k(i) for i in x2]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('k(x)')\nplt.grid()\n\n# Plot x against k(x)\nplt.plot(x1,y1, color='green')\nplt.plot(x2,y2, color='green')\n\n# plot a circle at the interval ends\nplt.plot(0, k(0), color='green', marker='o', markerfacecolor='green', markersize=10)\nplt.plot(0, k(0.0001), color='green', marker='o', markerfacecolor='w', markersize=10)\n\nplt.show()",
159      "execution_count": 7,
160      "outputs": [
161        {
162          "output_type": "display_data",
163          "data": {
164            "image/png": 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\n",
165            "text/plain": "<matplotlib.figure.Figure at 0x7fb64cf704e0>"
166          },
167          "metadata": {}
168        }
169      ]
170    },
171    {
172      "metadata": {},
173      "cell_type": "markdown",
174      "source": "\n## Finding Limits of Functions Graphically\nSo the question arises, how do we find a value for the limit of a function at a specific point?\n\nLet's explore this function, ***a***:\n\n\\begin{equation}a(x) = x^{2} + 1\\end{equation}\n\nWe can start by plotting it:"
175    },
176    {
177      "metadata": {
178        "trusted": true
179      },
180      "cell_type": "code",
181      "source": "%matplotlib inline\n\n# Define function a\ndef a(x):\n    return x**2 + 1\n\n\n# Plot output from function a\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-10, 11)\n\n# Get the corresponding y values from the function\ny = [a(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('a(x)')\nplt.grid()\n\n# Plot x against a(x)\nplt.plot(x,y, color='purple')\n\nplt.show()",
182      "execution_count": 8,
183      "outputs": [
184        {
185          "output_type": "display_data",
186          "data": {
187            "image/png": 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\n",
188            "text/plain": "<matplotlib.figure.Figure at 0x7fb64c5d16d8>"
189          },
190          "metadata": {}
191        }
192      ]
193    },
194    {
195      "metadata": {},
196      "cell_type": "markdown",
197      "source": "Note that this function is continuous at all points, there are no gaps in its range. However, the range of the function is *{a(x) &ge; 1}* (in other words, all real numbers that are greater than or equal to 1). For negative values of ***x***, the function appears to return ever-decreasing values as ***x*** gets closer to 0, and for positive values of ***x***, the function appears to return ever-increasing values as ***x*** gets further from 0; but it never returns 0.\n\nLet's plot the function for an ***x*** value of 0 and find out what the ***a(0)*** value is returned:"
198    },
199    {
200      "metadata": {
201        "trusted": true
202      },
203      "cell_type": "code",
204      "source": "%matplotlib inline\n\n# Define function a\ndef a(x):\n    return x**2 + 1\n\n\n# Plot output from function a\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-10, 11)\n\n# Get the corresponding y values from the function\ny = [a(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('a(x)')\nplt.grid()\n\n# Plot x against a(x)\nplt.plot(x,y, color='purple')\n\n# Plot a(x) when x = 0\nzx = 0\nzy = a(zx)\nplt.plot(zx, zy, color='red', marker='o', markersize=10)\nplt.annotate(str(zy),(zx, zy), xytext=(zx, zy + 5))\n\nplt.show()",
205      "execution_count": 9,
206      "outputs": [
207        {
208          "output_type": "display_data",
209          "data": {
210            "image/png": 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\n",
211            "text/plain": "<matplotlib.figure.Figure at 0x7fb64c5a71d0>"
212          },
213          "metadata": {}
214        }
215      ]
216    },
217    {
218      "metadata": {},
219      "cell_type": "markdown",
220      "source": "OK, so ***a(0)*** returns **1**.\n\nWhat happens if we use ***x*** values that are very slightly higher or lower than 0?"
221    },
222    {
223      "metadata": {
224        "trusted": true
225      },
226      "cell_type": "code",
227      "source": "%matplotlib inline\n\n# Define function a\ndef a(x):\n    return x**2 + 1\n\n\n# Plot output from function a\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-10, 11)\n\n# Get the corresponding y values from the function\ny = [a(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('a(x)')\nplt.grid()\n\n# Plot x against a(x)\nplt.plot(x,y, color='purple')\n\n# Plot a(x) when x = 0.1\nposx = 0.1\nposy = a(posx)\nplt.plot(posx, posy, color='blue', marker='<', markersize=10)\nplt.annotate(str(posy),(posx, posy), xytext=(posx + 1, posy))\n\n# Plot a(x) when x = -0.1\nnegx = -0.1\nnegy = a(negx)\nplt.plot(negx, negy, color='orange', marker='>', markersize=10)\nplt.annotate(str(negy),(negx, negy), xytext=(negx - 2, negy))\n\nplt.show()",
228      "execution_count": 10,
229      "outputs": [
230        {
231          "output_type": "display_data",
232          "data": {
233            "image/png": 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\n",
234            "text/plain": "<matplotlib.figure.Figure at 0x7fb64c5b8978>"
235          },
236          "metadata": {}
237        }
238      ]
239    },
240    {
241      "metadata": {},
242      "cell_type": "markdown",
243      "source": "These ***x*** values return ***a(x)*** values that are just slightly above 1, and if we were to keep plotting numbers that are increasingly close to 0, for example 0.0000000001 or -0.0000000001, the function would still return a value that is just slightly greater than 1. The limit of function *a(x)* as *x* approaches 0, is 1; and the notation to indicate this is:\n\n\\begin{equation}\\lim_{x \\to 0} a(x) = 1 \\end{equation}\n\nThis reflects a more formal definition of function continuity. Previously, we stated that a function is continuous at a point if you can draw it at that point without lifting your pen. The more mathematical definition is that a function is continuous at a point if the limit of the function as it approaches that point from both directions is equal to the function's value at that point. In this case, as we approach x = 0 from both sides, the limit is 1; and the value of *a(0)* is also 1; so the function is continuous at x = 0."
244    },
245    {
246      "metadata": {},
247      "cell_type": "markdown",
248      "source": "### Limits at Non-Continuous Points\nLet's try another function, which we'll call ***b***:\n\n\\begin{equation}b(x) = -2x^{2} \\cdot \\frac{1}{x},\\;\\;x\\ne0\\end{equation}\n\nNote that this function has a domain that includes all real number values of *x* such that *x* does not equal 0. In other words, the function will return a valid output for any number other than 0.\n\nLet's create it and plot it with Python:"
249    },
250    {
251      "metadata": {
252        "trusted": true
253      },
254      "cell_type": "code",
255      "source": "%matplotlib inline\n\n# Define function b\ndef b(x):\n    if x != 0:\n        return (-2*x**2) * 1/x\n\n\n# Plot output from function g\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-10, 11)\n\n# Get the corresponding y values from the function\ny = [b(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('b(x)')\nplt.grid()\n\n# Plot x against b(x)\nplt.plot(x,y, color='purple')\n\nplt.show()",
256      "execution_count": 11,
257      "outputs": [
258        {
259          "output_type": "display_data",
260          "data": {
261            "image/png": 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\n",
262            "text/plain": "<matplotlib.figure.Figure at 0x7fb64c5fb630>"
263          },
264          "metadata": {}
265        }
266      ]
267    },
268    {
269      "metadata": {},
270      "cell_type": "markdown",
271      "source": "The output from this function contains a gap in the line where x = 0. It seems that not only does the *domain* of the function (the values that can be passed in as *x*) exclude 0; but the *range* of the function (the set of values that can be returned from it) also excludes 0.\n\nWe can't evaluate the function for an *x* value of 0, but we can see what it returns for a value that is just very slightly less than 0:"
272    },
273    {
274      "metadata": {
275        "collapsed": true,
276        "trusted": false
277      },
278      "cell_type": "code",
279      "source": "%matplotlib inline\n\n# Define function b\ndef b(x):\n    if x != 0:\n        return (-2*x**2) * 1/x\n\n\n# Plot output from function g\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-10, 11)\n\n# Get the corresponding y values from the function\ny = [b(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('b(x)')\nplt.grid()\n\n# Plot x against b(x)\nplt.plot(x,y, color='purple')\n\n# Plot b(x) for x = -0.1\nnegx = -0.1\nnegy = b(negx)\nplt.plot(negx, negy, color='orange', marker='>', markersize=10)\nplt.annotate(str(negy),(negx, negy), xytext=(negx + 1, negy))\n\nplt.show()",
280      "execution_count": null,
281      "outputs": []
282    },
283    {
284      "metadata": {},
285      "cell_type": "markdown",
286      "source": "We can even try a negative *x* value that's a little closer to 0."
287    },
288    {
289      "metadata": {
290        "trusted": true
291      },
292      "cell_type": "code",
293      "source": "%matplotlib inline\n\n# Define function b\ndef b(x):\n    if x != 0:\n        return (-2*x**2) * 1/x\n\n\n# Plot output from function g\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-10, 11)\n\n# Get the corresponding y values from the function\ny = [b(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('b(x)')\nplt.grid()\n\n# Plot x against b(x)\nplt.plot(x,y, color='purple')\n\n# Plot b(x) for x = -0.0001\nnegx = -0.0001\nnegy = b(negx)\nplt.plot(negx, negy, color='orange', marker='>', markersize=10)\nplt.annotate(str(negy),(negx, negy), xytext=(negx + 1, negy))\n\nplt.show()",
294      "execution_count": 12,
295      "outputs": [
296        {
297          "output_type": "display_data",
298          "data": {
299            "image/png": 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\n",
300            "text/plain": "<matplotlib.figure.Figure at 0x7fb64c5b8a20>"
301          },
302          "metadata": {}
303        }
304      ]
305    },
306    {
307      "metadata": {},
308      "cell_type": "markdown",
309      "source": "So as the value of *x* gets closer to 0 from the left (negative), the value of *b(x)* is decreasing towards 0. We can show this with the following notation:\n\n\\begin{equation}\\lim_{x \\to 0^{-}} b(x) = 0 \\end{equation}\n\nNote that the arrow points to 0<sup>-</sup> (with a minus sign) to indicate that we're describing the limit as we approach 0 from the negative side.\n\nSo what about the positive side?\n\nLet's see what the function value is when *x* is 0.1:"
310    },
311    {
312      "metadata": {
313        "trusted": true
314      },
315      "cell_type": "code",
316      "source": "%matplotlib inline\n\n# Define function b\ndef b(x):\n    if x != 0:\n        return (-2*x**2) * 1/x\n\n\n# Plot output from function g\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-10, 11)\n\n# Get the corresponding y values from the function\ny = [b(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('b(x)')\nplt.grid()\n\n# Plot x against b(x)\nplt.plot(x,y, color='purple')\n\n# Plot b(x) for x = 0.1\nposx = 0.1\nposy = b(posx)\nplt.plot(posx, posy, color='blue', marker='<', markersize=10)\nplt.annotate(str(posy),(posx, posy), xytext=(posx + 1, posy))\n\nplt.show()",
317      "execution_count": 13,
318      "outputs": [
319        {
320          "output_type": "display_data",
321          "data": {
322            "image/png": 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\n",
323            "text/plain": "<matplotlib.figure.Figure at 0x7fb64d266898>"
324          },
325          "metadata": {}
326        }
327      ]
328    },
329    {
330      "metadata": {},
331      "cell_type": "markdown",
332      "source": "What happens if we decrease the value of *x* so that it's even closer to 0?"
333    },
334    {
335      "metadata": {
336        "trusted": true
337      },
338      "cell_type": "code",
339      "source": "%matplotlib inline\n\n# Define function b\ndef b(x):\n    if x != 0:\n        return (-2*x**2) * 1/x\n\n\n# Plot output from function g\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-10, 11)\n\n# Get the corresponding y values from the function\ny = [b(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('b(x)')\nplt.grid()\n\n# Plot x against b(x)\nplt.plot(x,y, color='purple')\n\n# Plot b(x) for x = 0.0001\nposx = 0.0001\nposy = b(posx)\nplt.plot(posx, posy, color='blue', marker='<', markersize=10)\nplt.annotate(str(posy),(posx, posy), xytext=(posx + 1, posy))\n\nplt.show()",
340      "execution_count": 14,
341      "outputs": [
342        {
343          "output_type": "display_data",
344          "data": {
345            "image/png": 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\n",
346            "text/plain": "<matplotlib.figure.Figure at 0x7fb64d277860>"
347          },
348          "metadata": {}
349        }
350      ]
351    },
352    {
353      "metadata": {},
354      "cell_type": "markdown",
355      "source": "As with the negative side, as *x* approaches 0 from the positive side, the value of *b(x)* gets closer to 0; and we can show that like this:\n\n\\begin{equation}\\lim_{x \\to 0^{+}} b(x) = 0 \\end{equation}\n\nNow, even although the function is not defined at x = 0; since the limit as we approach x = 0 from the negative side is 0, and the limit when we approach x = 0 from the positive side is also 0; we can say that the overall, or *two-sided* limit for the function at x = 0 is 0:\n\n\\begin{equation}\\lim_{x \\to 0} b(x) = 0 \\end{equation}\n\nSo can we therefore just ignore the gap and say that the function is *continuous* at x = 0? Well, recall that the formal definition for continuity is that to be continuous at a point, the function's limit as we approach the point in both directions must be equal to the function's value at that point. In this case, the two-sided limit as we approach x = 0 is 0, but *b(0)* is not defined; so the function is ***non-continuous*** at x = 0."
356    },
357    {
358      "metadata": {},
359      "cell_type": "markdown",
360      "source": "### One-Sided Limits\nLet's take a look at a different function. We'll call this one ***c***:\n\n\\begin{equation}\nc(x) = \\begin{cases}\n  x + 20, & \\text{if } x \\le 0, \\\\\n  x - 100, & \\text{otherwise }\n\\end{cases}\n\\end{equation}\n\nIn this case, the function's domain includes all real numbers, but its range is still non-continuous because of the way different values are returned depending on the value of *x*. The range of possible outputs for *c(x &le; 0)* is &le; 20, and the range of output values for *c(x > 0)* is x &ge; -100.\n\nLet's use Python to plot function ***c*** with some values for *c(x)* marked on the line"
361    },
362    {
363      "metadata": {
364        "trusted": true
365      },
366      "cell_type": "code",
367      "source": "%matplotlib inline\n\ndef c(x):\n    import numpy as np\n    if x <= 0:\n        return x + 20\n    else:\n        return x - 100\n\n# Plot output from function h\nfrom matplotlib import pyplot as plt\n\n# Create arrays of x values\nx1 = range(-20, 6)\nx2 = range(6, 21)\n\n# Get the corresponding y values from the function\ny1 = [c(i) for i in x1]\ny2 = [c(i) for i in x2]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('c(x)')\nplt.grid()\n\n# Plot x against c(x)\nplt.plot(x1,y1, color='purple')\nplt.plot(x2,y2, color='purple')\n\n# plot a circle close enough to the c limits for our purposes!\nplt.plot(5, c(5), color='purple', marker='o', markerfacecolor='purple', markersize=10)\nplt.plot(5, c(5.001), color='purple', marker='o', markerfacecolor='w', markersize=10)\n\n# plot some points from the +ve direction\nposx = [20, 15, 10, 6]\nposy = [c(i) for i in posx]\nplt.scatter(posx, posy, color='blue', marker='<', s=70)\nfor p in posx:\n    plt.annotate(str(c(p)),(p, c(p)),xytext=(p, c(p) + 5))\n    \n# plot some points from the -ve direction\nnegx = [-15, -10, -5, 0, 4]\nnegy = [c(i) for i in negx]\nplt.scatter(negx, negy, color='orange', marker='>', s=70)\nfor n in negx:\n    plt.annotate(str(c(n)),(n, c(n)),xytext=(n, c(n) + 5))\n\nplt.show()",
368      "execution_count": 15,
369      "outputs": [
370        {
371          "output_type": "display_data",
372          "data": {
373            "image/png": 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\n",
374            "text/plain": "<matplotlib.figure.Figure at 0x7fb64cf70438>"
375          },
376          "metadata": {}
377        }
378      ]
379    },
380    {
381      "metadata": {},
382      "cell_type": "markdown",
383      "source": "The plot of the function shows a line in which the *c(x)* value increases towards 25 as *x* approaches 5 from the negative side:\n\n\\begin{equation}\\lim_{x \\to 5^{-}} c(x) = 25 \\end{equation}\n\nHowever, the *c(x)* value decreases towards -95 as *x* approaches 5 from the positive side:\n\n\\begin{equation}\\lim_{x \\to 5^{+}} c(x) = -95 \\end{equation}\n\nSo what can we say about the two-sided limit of this function at x = 5?\n\nThe limit as we approach x = 5 from the negative side is *not* equal to the limit as we approach x = 5 from the positive side, so no two-sided limit exists for this function at that point:\n\n\\begin{equation}\\lim_{x \\to 5} \\text{does not exist} \\end{equation}"
384    },
385    {
386      "metadata": {},
387      "cell_type": "markdown",
388      "source": "### Asymptotes and Infinity\nOK, time to look at another function:\n\n\\begin{equation}d(x) = \\frac{4}{x - 25},\\;\\; x \\ne 25\\end{equation}"
389    },
390    {
391      "metadata": {
392        "trusted": true
393      },
394      "cell_type": "code",
395      "source": "%matplotlib inline\n\n# Define function d\ndef d(x):\n    if x != 25:\n        return 4 / (x - 25)\n\n\n# Plot output from function d\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = list(range(-100, 24))\nx.append(24.9) # Add some fractional x\nx.append(25)   # values around\nx.append(25.1) # 25 for finer-grain results\nx = x + list(range(26, 101))\n# Get the corresponding y values from the function\ny = [d(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('d(x)')\nplt.grid()\n\n# Plot x against d(x)\nplt.plot(x,y, color='purple')\n\nplt.show()",
396      "execution_count": 16,
397      "outputs": [
398        {
399          "output_type": "display_data",
400          "data": {
401            "image/png": 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\n",
402            "text/plain": "<matplotlib.figure.Figure at 0x7fb64eae07f0>"
403          },
404          "metadata": {}
405        }
406      ]
407    },
408    {
409      "metadata": {},
410      "cell_type": "markdown",
411      "source": "What's the limit of *d* as *x* approaches 25?\n\nWe can plot a few points to help us:"
412    },
413    {
414      "metadata": {
415        "trusted": true
416      },
417      "cell_type": "code",
418      "source": "%matplotlib inline\n\n# Define function d\ndef d(x):\n    if x != 25:\n        return 4 / (x - 25)\n\n\n# Plot output from function d\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = list(range(-100, 24))\nx.append(24.9) # Add some fractional x\nx.append(25)   # values around\nx.append(25.1) # 25 for finer-grain results\nx = x + list(range(26, 101))\n# Get the corresponding y values from the function\ny = [d(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('d(x)')\nplt.grid()\n\n# Plot x against d(x)\nplt.plot(x,y, color='purple')\n\n# plot some points from the +ve direction\nposx = [75, 50, 30, 25.5, 25.2, 25.1]\nposy = [d(i) for i in posx]\nplt.scatter(posx, posy, color='blue', marker='<')\nfor p in posx:\n    plt.annotate(str(d(p)),(p, d(p)))\n    \n# plot some points from the -ve direction\nnegx = [-55, 0, 23, 24.5, 24.8, 24.9]\nnegy = [d(i) for i in negx]\nplt.scatter(negx, negy, color='orange', marker='>')\nfor n in negx:\n    plt.annotate(str(d(n)),(n, d(n)))\n\nplt.show()",
419      "execution_count": 17,
420      "outputs": [
421        {
422          "output_type": "display_data",
423          "data": {
424            "image/png": 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\n",
425            "text/plain": "<matplotlib.figure.Figure at 0x7fb64eae05f8>"
426          },
427          "metadata": {}
428        }
429      ]
430    },
431    {
432      "metadata": {
433        "collapsed": true
434      },
435      "cell_type": "markdown",
436      "source": "From these plotted values, we can see that as *x* approaches 25 from the negative side, *d(x)* is decreasing, and as *x* approaches 25 from the positive side, *d(x)* is increasing. As *x* gets closer to 25, *d(x)* increases or decreases more significantly.\n\nIf we were to plot every fractional value of *d(x)* for *x* values between 24.9 and 25, we'd see a line that decreases indefintely, getting closer and closer to the x = 25 vertical line, but never actually reaching it. Similarly, plotting every *x* value between 25 and 25.1 would result in a line going up indefinitely, but always staying to the right of the vertical x = 25 line.\n\nThe x = 25 line in this case is an *asymptote* - a line to which a curve moves ever closer but never actually reaches. The positive limit for x = 25 in this case in not a real numbered value, but *infinity*:\n\n\\begin{equation}\\lim_{x \\to 25^{+}} d(x) = \\infty \\end{equation}\n\nConversely, the negative limit for x = 25 is negative infinity:\n\n\\begin{equation}\\lim_{x \\to 25^{-}} d(x) = -\\infty \\end{equation}\n\n"
437    },
438    {
439      "metadata": {},
440      "cell_type": "markdown",
441      "source": "## Finding Limits Numerically Using a Table\nUp to now, we've estimated limits for a point graphically by examining a graph of a function. You can also approximate limits by creating a table of x values and the corresponding function values either side of the point for which you want to find the limits.\n\nFor example, let's return to our ***a*** function:\n\n\\begin{equation}a(x) = x^{2} + 1\\end{equation}\n\nIf we want to find the limits as x is approaching 0, we can apply the function to some values either side of 0 and view them as a table. Here's some Python code to do that:"
442    },
443    {
444      "metadata": {
445        "trusted": true
446      },
447      "cell_type": "code",
448      "source": "# Define function a\ndef a(x):\n    return x**2 + 1\n\n\nimport pandas as pd\n\n# Create a dataframe with an x column containing values either side of 0\ndf = pd.DataFrame ({'x': [-1, -0.5, -0.2, -0.1, -0.01, 0, 0.01, 0.1, 0.2, 0.5, 1]})\n\n# Add an a(x) column by applying the function to x\ndf['a(x)'] = a(df['x'])\n\n#Display the dataframe\ndf",
449      "execution_count": 18,
450      "outputs": [
451        {
452          "output_type": "execute_result",
453          "execution_count": 18,
454          "data": {
455            "text/html": "<div>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>x</th>\n      <th>a(x)</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>0</th>\n      <td>-1.00</td>\n      <td>2.0000</td>\n    </tr>\n    <tr>\n      <th>1</th>\n      <td>-0.50</td>\n      <td>1.2500</td>\n    </tr>\n    <tr>\n      <th>2</th>\n      <td>-0.20</td>\n      <td>1.0400</td>\n    </tr>\n    <tr>\n      <th>3</th>\n      <td>-0.10</td>\n      <td>1.0100</td>\n    </tr>\n    <tr>\n      <th>4</th>\n      <td>-0.01</td>\n      <td>1.0001</td>\n    </tr>\n    <tr>\n      <th>5</th>\n      <td>0.00</td>\n      <td>1.0000</td>\n    </tr>\n    <tr>\n      <th>6</th>\n      <td>0.01</td>\n      <td>1.0001</td>\n    </tr>\n    <tr>\n      <th>7</th>\n      <td>0.10</td>\n      <td>1.0100</td>\n    </tr>\n    <tr>\n      <th>8</th>\n      <td>0.20</td>\n      <td>1.0400</td>\n    </tr>\n    <tr>\n      <th>9</th>\n      <td>0.50</td>\n      <td>1.2500</td>\n    </tr>\n    <tr>\n      <th>10</th>\n      <td>1.00</td>\n      <td>2.0000</td>\n    </tr>\n  </tbody>\n</table>\n</div>",
456            "text/plain": "       x    a(x)\n0  -1.00  2.0000\n1  -0.50  1.2500\n2  -0.20  1.0400\n3  -0.10  1.0100\n4  -0.01  1.0001\n5   0.00  1.0000\n6   0.01  1.0001\n7   0.10  1.0100\n8   0.20  1.0400\n9   0.50  1.2500\n10  1.00  2.0000"
457          },
458          "metadata": {}
459        }
460      ]
461    },
462    {
463      "metadata": {},
464      "cell_type": "markdown",
465      "source": "Looking at the output, you can see that the function values are getting closer to 1 as x approaches 0 from both sides, so:\n\n\\begin{equation}\\lim_{x \\to 0} a(x) = 1 \\end{equation}\n\nAdditionally, you can see that the actual value of the function when x = 0 is also 1, so:\n\n\\begin{equation}\\lim_{x \\to 0} a(x) = a(0) \\end{equation}\n\nWhich according to our earlier definition, means that the function is continuous at 0.\n\nHowever, you should be careful not to assume that the limit when x is approaching 0 will always be the same as the value when x = 0; even when the function is defined for x = 0.\n\nFor example, consider the following function:\n\n\\begin{equation}\ne(x) = \\begin{cases}\n  5, & \\text{if } x = 0, \\\\\n  1 + x^{2}, & \\text{otherwise }\n\\end{cases}\n\\end{equation}\n\nLet's see what the function returns for *x* values either side of 0 in a table:"
466    },
467    {
468      "metadata": {
469        "trusted": true
470      },
471      "cell_type": "code",
472      "source": "# Define function e\ndef e(x):\n    if x == 0:\n        return 5\n    else:\n        return 1 + x**2\n\nimport pandas as pd\n# Create a dataframe with an x column containing values either side of 0\nx= [-1, -0.5, -0.2, -0.1, -0.01, 0, 0.01, 0.1, 0.2, 0.5, 1]\ny =[e(i) for i in x]\ndf = pd.DataFrame ({' x':x, 'e(x)': y })\ndf",
473      "execution_count": 19,
474      "outputs": [
475        {
476          "output_type": "execute_result",
477          "execution_count": 19,
478          "data": {
479            "text/html": "<div>\n<table border=\"1\" class=\"dataframe\">\n  <thead>\n    <tr style=\"text-align: right;\">\n      <th></th>\n      <th>x</th>\n      <th>e(x)</th>\n    </tr>\n  </thead>\n  <tbody>\n    <tr>\n      <th>0</th>\n      <td>-1.00</td>\n      <td>2.0000</td>\n    </tr>\n    <tr>\n      <th>1</th>\n      <td>-0.50</td>\n      <td>1.2500</td>\n    </tr>\n    <tr>\n      <th>2</th>\n      <td>-0.20</td>\n      <td>1.0400</td>\n    </tr>\n    <tr>\n      <th>3</th>\n      <td>-0.10</td>\n      <td>1.0100</td>\n    </tr>\n    <tr>\n      <th>4</th>\n      <td>-0.01</td>\n      <td>1.0001</td>\n    </tr>\n    <tr>\n      <th>5</th>\n      <td>0.00</td>\n      <td>5.0000</td>\n    </tr>\n    <tr>\n      <th>6</th>\n      <td>0.01</td>\n      <td>1.0001</td>\n    </tr>\n    <tr>\n      <th>7</th>\n      <td>0.10</td>\n      <td>1.0100</td>\n    </tr>\n    <tr>\n      <th>8</th>\n      <td>0.20</td>\n      <td>1.0400</td>\n    </tr>\n    <tr>\n      <th>9</th>\n      <td>0.50</td>\n      <td>1.2500</td>\n    </tr>\n    <tr>\n      <th>10</th>\n      <td>1.00</td>\n      <td>2.0000</td>\n    </tr>\n  </tbody>\n</table>\n</div>",
480            "text/plain": "       x    e(x)\n0  -1.00  2.0000\n1  -0.50  1.2500\n2  -0.20  1.0400\n3  -0.10  1.0100\n4  -0.01  1.0001\n5   0.00  5.0000\n6   0.01  1.0001\n7   0.10  1.0100\n8   0.20  1.0400\n9   0.50  1.2500\n10  1.00  2.0000"
481          },
482          "metadata": {}
483        }
484      ]
485    },
486    {
487      "metadata": {},
488      "cell_type": "markdown",
489      "source": "As before, you can see that as the *x* values approach 0 from both sides, the value of the function gets closer to 1, so:\n\n\\begin{equation}\\lim_{x \\to 0} e(x) = 1 \\end{equation}\n\nHowever the actual value of the function when x = 0 is 5, not 1; so:\n\n\\begin{equation}\\lim_{x \\to 0} e(x) \\ne e(0) \\end{equation}\n\nWhich according to our earlier definition, means that the function is non-continuous at 0.\n\nRun the following cell to see what this looks like as a graph:"
490    },
491    {
492      "metadata": {
493        "trusted": true
494      },
495      "cell_type": "code",
496      "source": "%matplotlib inline\n\n# Define function e\ndef e(x):\n    if x == 0:\n        return 5\n    else:\n        return 1 + x**2\n\nfrom matplotlib import pyplot as plt\n\nx= [-1, -0.5, -0.2, -0.1, -0.01, 0.01, 0.1, 0.2, 0.5, 1]\ny =[e(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('e(x)')\nplt.grid()\n\n# Plot x against e(x)\nplt.plot(x, y, color='purple')\n# (we're cheating slightly - we'll manually plot the discontinous point...)\nplt.scatter(0, e(0), color='purple')\n# (... and overplot the gap)\nplt.plot(0, 1, color='purple', marker='o', markerfacecolor='w', markersize=10)\nplt.show()",
497      "execution_count": 20,
498      "outputs": [
499        {
500          "output_type": "display_data",
501          "data": {
502            "image/png": 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\n",
503            "text/plain": "<matplotlib.figure.Figure at 0x7fb64d266da0>"
504          },
505          "metadata": {}
506        }
507      ]
508    },
509    {
510      "metadata": {
511        "collapsed": true
512      },
513      "cell_type": "markdown",
514      "source": "## Determining Limits Analytically\nWe've seen how to estimate limits visually on a graph, and by creating a table of *x* and *f(x)* values either side of a point. There are also some mathematical techniques we can use to calculate limits.\n\n### Direct Substitution\nRecall that our definition for a function to be continuous at a point is that the two-directional limit must exist and that it must be equal to the function value at that point. It therefore follows, that if we know that a function is continuous at a given point, we can determine the limit simply by evaluating the function for that point.\n\nFor example, let's consider the following function ***g***:\n\n\\begin{equation}g(x) = \\frac{x^{2} - 1}{x - 1}, x \\ne 1\\end{equation}\n\nRun the following code to see this function as a graph:"
515    },
516    {
517      "metadata": {
518        "trusted": true
519      },
520      "cell_type": "code",
521      "source": "%matplotlib inline\n\n# Define function f\ndef g(x):\n    if x != 1:\n        return (x**2 - 1) / (x - 1)\n\n\n# Plot output from function g\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx= range(-20, 21)\ny =[g(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('g(x)')\nplt.grid()\n\n# Plot x against g(x)\nplt.plot(x,y, color='purple')\n\nplt.show()",
522      "execution_count": 21,
523      "outputs": [
524        {
525          "output_type": "display_data",
526          "data": {
527            "image/png": 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\n",
528            "text/plain": "<matplotlib.figure.Figure at 0x7fb64d26f080>"
529          },
530          "metadata": {}
531        }
532      ]
533    },
534    {
535      "metadata": {},
536      "cell_type": "markdown",
537      "source": "Now, suppose we need to find the limit of ***g(x)*** as ***x*** approaches **4**. We can try to find this by simply substituting 4 for the *x* values in the function:\n\n\\begin{equation}g(4) = \\frac{4^{2} - 1}{4 - 1}\\end{equation}\n\nThis simplifies to:\n\n\\begin{equation}g(4) = \\frac{15}{3}\\end{equation}\n\nSo:\n\n\\begin{equation}\\lim_{x \\to 4} g(x) = 5\\end{equation}\n\nLet's take a look:"
538    },
539    {
540      "metadata": {
541        "trusted": true
542      },
543      "cell_type": "code",
544      "source": "%matplotlib inline\n\n# Define function g\ndef g(x):\n    if x != 1:\n        return (x**2 - 1) / (x - 1)\n\n\n# Plot output from function f\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx= range(-20, 21)\ny =[g(i) for i in x]\n\n# Set the x point we're interested in\nzx = 4\n\nplt.xlabel('x')\nplt.ylabel('g(x)')\nplt.grid()\n\n# Plot x against g(x)\nplt.plot(x,y, color='purple')\n\n# Plot g(x) when x = 0\nzy = g(zx)\nplt.plot(zx, zy, color='red', marker='o', markersize=10)\nplt.annotate(str(zy),(zx, zy), xytext=(zx - 2, zy + 1))\n\nplt.show()\n\nprint ('Limit as x -> ' + str(zx) + ' = ' + str(zy))",
545      "execution_count": 22,
546      "outputs": [
547        {
548          "output_type": "display_data",
549          "data": {
550            "image/png": 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\n",
551            "text/plain": "<matplotlib.figure.Figure at 0x7fb64c582048>"
552          },
553          "metadata": {}
554        },
555        {
556          "output_type": "stream",
557          "text": "Limit as x -> 4 = 5.0\n",
558          "name": "stdout"
559        }
560      ]
561    },
562    {
563      "metadata": {},
564      "cell_type": "markdown",
565      "source": "### Factorization\nOK, now let's try to find the limit of ***g(x)*** as ***x*** approaches **1**.\n\nWe know from the function definition that the function is not defined at x = 1, but we're not trying to find the *value* of ***g(x)*** when x *equals* 1; we're trying to find the *limit* of ***g(x)*** as x *approaches* 1.\n\nThe direct substitution approach won't work in this case:\n\n\\begin{equation}g(1) = \\frac{1^{2} - 1}{1 - 1}\\end{equation}\n\nSimplifies to:\n\n\\begin{equation}g(1) = \\frac{0}{0}\\end{equation}\n\nAnything divided by 0 is undefined; so all we've done is to confirm that the function is not defined at this point. You might be tempted to assume that this means the limit does not exist, but <sup>0</sup>/<sub>0</sub> is a special case; it's what's known as the *indeterminate form*; and there may be a way to solve this problem another way.\n\nWe can factor the *x<sup>2</sup> - 1* numerator in the definition of ***g*** as as *(x - 1)(x + 1)*, so the limit equation can we rewritten like this:\n\n\\begin{equation}\\lim_{x \\to a} g(x) = \\frac{(x-1)(x+1)}{x - 1}\\end{equation}\n\nThe ***x - 1*** in the numerator and the ***x - 1*** in the denominator cancel each other out:\n\n\\begin{equation}\\lim_{x \\to a} g(x)= x+1\\end{equation}\n\nSo we can now use substitution for *x = 1* to calculate the limit as *1 + 1*:\n\n\\begin{equation}\\lim_{x \\to 1} g(x) = 2\\end{equation}\n\nLet's see what that looks like:"
566    },
567    {
568      "metadata": {
569        "trusted": true
570      },
571      "cell_type": "code",
572      "source": "%matplotlib inline\n\n# Define function g\ndef f(x):\n    if x != 1:\n        return (x**2 - 1) / (x - 1)\n\n\n# Plot output from function g\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx= range(-20, 21)\ny =[g(i) for i in x]\n\n# Set the x point we're interested in\nzx = 1\n\n# Calculate the limit of g(x) when x->zx using the factored equation\nzy = zx + 1\n\nplt.xlabel('x')\nplt.ylabel('g(x)')\nplt.grid()\n\n# Plot x against g(x)\nplt.plot(x,y, color='purple')\n\n# Plot the limit of g(x)\nzy = zx + 1\nplt.plot(zx, zy, color='red', marker='o', markersize=10)\nplt.annotate(str(zy),(zx, zy), xytext=(zx - 2, zy + 1))\n\nplt.show()\n\nprint ('Limit as x -> ' + str(zx) + ' = ' + str(zy))",
573      "execution_count": 23,
574      "outputs": [
575        {
576          "output_type": "display_data",
577          "data": {
578            "image/png": 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\n",
579            "text/plain": "<matplotlib.figure.Figure at 0x7fb64c57f908>"
580          },
581          "metadata": {}
582        },
583        {
584          "output_type": "stream",
585          "text": "Limit as x -> 1 = 2\n",
586          "name": "stdout"
587        }
588      ]
589    },
590    {
591      "metadata": {},
592      "cell_type": "markdown",
593      "source": "### Rationalization\nLet's look at another function:\n\n\\begin{equation}h(x) = \\frac{\\sqrt{x} - 2}{x - 4}, x \\ne 4 \\text{ and } x \\ge 0\\end{equation}\n\nRun the following cell to plot this function as a graph:"
594    },
595    {
596      "metadata": {
597        "trusted": true
598      },
599      "cell_type": "code",
600      "source": "%matplotlib inline\n\n# Define function h\ndef h(x):\n    import math\n    if x >= 0 and x != 4:\n        return (math.sqrt(x) - 2) / (x - 4)\n\n\n# Plot output from function h\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx= range(-20, 21)\ny =[h(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.ylabel('h(x)')\nplt.grid()\n\n# Plot x against h(x)\nplt.plot(x,y, color='purple')\n\nplt.show()",
601      "execution_count": 24,
602      "outputs": [
603        {
604          "output_type": "display_data",
605          "data": {
606            "image/png": 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\n",
607            "text/plain": "<matplotlib.figure.Figure at 0x7fb64c57f978>"
608          },
609          "metadata": {}
610        }
611      ]
612    },
613    {
614      "metadata": {},
615      "cell_type": "markdown",
616      "source": "To find the limit of ***h(x)*** as ***x*** approaches **4**, we can't use the direct substitution method because the function is not defined at that point. However, we can take an alternative approach by multiplying both the numerator and denominator in the function by the *conjugate* of the numerator to *rationalize* the square root term (a conjugate is a binomial formed by reversing the sign of the second term of a binomial):\n\n\\begin{equation}\\lim_{x \\to a}h(x) = \\frac{\\sqrt{x} - 2}{x - 4}\\cdot\\frac{\\sqrt{x} + 2}{\\sqrt{x} + 2}\\end{equation}\n\nThis simplifies to:\n\n\\begin{equation}\\lim_{x \\to a}h(x) = \\frac{(\\sqrt{x})^{2} - 2^{2}}{(x - 4)({\\sqrt{x} + 2})}\\end{equation}\n\nThe &radic;x<sup>2</sup> is x, and 2<sup>2</sup> is 4, so we can simplify the numerator as follows:\n\n\\begin{equation}\\lim_{x \\to a}h(x) = \\frac{x - 4}{(x - 4)({\\sqrt{x} + 2})}\\end{equation}\n\nNow we can cancel out the *x - 4* in both the numerator and denominator:\n\n\\begin{equation}\\lim_{x \\to a}h(x) = \\frac{1}{{\\sqrt{x} + 2}}\\end{equation}\n\nSo for x approaching 4, this is:\n\n\\begin{equation}\\lim_{x \\to 4}h(x) = \\frac{1}{{\\sqrt{4} + 2}}\\end{equation}\n\nThis simplifies to:\n\n\\begin{equation}\\lim_{x \\to 4}h(x) = \\frac{1}{2 + 2}\\end{equation}\n\nWhich is of course:\n\n\\begin{equation}\\lim_{x \\to 4}h(x) = \\frac{1}{4}\\end{equation}\n\nSo the limit of ***h(x)*** as ***x*** approaches **4** is <sup>1</sup>/<sub>4</sub> or 0.25.\n\nLet's calculate and plot this with Python:"
617    },
618    {
619      "metadata": {
620        "trusted": true
621      },
622      "cell_type": "code",
623      "source": "%matplotlib inline\n\n# Define function h\ndef h(x):\n    import math\n    if x >= 0 and x != 4:\n        return (math.sqrt(x) - 2) / (x - 4)\n\n\n# Plot output from function h\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx= range(-20, 21)\ny =[h(i) for i in x]\n\n# Specify the point we're interested in\nzx = 4\n\n# Calculate the limit of f(x) when x->zx using factored equation\nimport math\nzy = 1 / ((math.sqrt(zx)) + 2)\n\nplt.xlabel('x')\nplt.ylabel('h(x)')\nplt.grid()\n\n# Plot x against h(x)\nplt.plot(x,y, color='purple')\n\n# Plot the limit of h(x) when x->zx                                    \nplt.plot(zx, zy, color='red', marker='o', markersize=10)\nplt.annotate(str(zy),(zx, zy), xytext=(zx + 2, zy))\n\nplt.show()\n\nprint ('Limit as x -> ' + str(zx) + ' = ' + str(zy))",
624      "execution_count": 25,
625      "outputs": [
626        {
627          "output_type": "display_data",
628          "data": {
629            "image/png": 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J3UFEpJdoz0V5d5tZLnADcCMwGHgfWAv8xN1fCzZi73LJVy9h7bNrmXfXPIZPHs6A/D5/DoCInAbadczC3fcTHvJjAfBXYDlwBLgyuGi9U3K/ZGbMmcH7e99n/hfnJzqOiEiXiOcA9/PAPxG+90R95KEjuS0YPGEwl957KSueXMGGFzYkOo6ISKfFcz+LYnefGliS08xl/3oZ6363jhc+/QKfXf1Z0gamJTqSiEiHxbNl8bqZjQ8syWkmOTWZ6XOmU7+znj99+U+JjiMi0intOXV2pZmtAC4F3jSz9Wa2Imq+tKLogiI+8JUPsOyJZVQtqEp0HBGRDmvPbqgPB57iNHb5Ny5n3XPr+P0dv+czqz5D/6z+iY4kIhK3mFsW7r6lrUd3hOzN+qX3Y8acGdRuq+Wle15KdBwRkQ4J9D7aZjY1sttqk5nd08LyOyO7s5ab2atmNi5q2b2RfuvN7JogcwZt2AeGceEXLqTyR5VsfmVzouOIiMQtsGJhZsnAo8A0YBzwsehiEPGUu4939wnAt4GHI33HATOBs4CpwI8ir9drXfGtK8gtzaXiUxUcO3Qs0XFEROIS5JbFJGCTu1e7+1FgLs1uPtpsvKkMoHF87xnAXHc/4u5vA5sir9drpWakMv2J6eyv2s/L//pyouOIiMQlyGJRBGyLmq6JzDuJmd1lZlWEtyw+H0/f3mb45OGUf6acxf+9mG2vb4vdQUSkh7CgbtZjZjcC17j7HZHpWwgPc/65VtrfFGl/q5k9CvzN3Z+MLHsCmOfuzzbrMwuYBVBYWDhx7ty5Hc5bX19PZmZmh/u3V8OhBipvryQpNYnyx8tJSm27XndXrngpV3yUKz7KFZ/O5JoyZcpSdy+P2dDdA3kAFwPzo6bvBe5to30SUNtSW2A+cHFb7zdx4kTvjIULF3aqfzw2zd/k93O/L7hnQcy23ZkrHsoVH+WKj3LFpzO5gEpvx3d6kLuhlgCjzWyEmaUSPmBdEd0gcqvWRtcRvvsekXYzzay/mY0ARgNvBJi1W428eiQTbp/A6995nR2VOxIdR0QkpsCKhbs3ALMJbxWsBZ5299Vm9oCZTY80m21mq81sOfAvwK2RvquBp4E1wB+Bu9z9eFBZE+Ga715DZmEmz970LHU1p9xXSkSkRwn0Ogt3n+fuZ7r7SHd/MDLvPneviDz/gruf5e4T3H1KpEg09n0w0m+Mu78YZM5ESBuYxg1P30D9znrmXDqHfZv2JTqSiEirAi0W0raSS0q4deGtHHvvGHMuncO7K99NdCQRkRapWCTY0IlDue0vt5GUksTPLv8ZNYtrEh1JROQUKhY9QEFZAbe/ejvpeen84qpfUP3n6kRHEhE5iYpFDzFw+EA++ddPkjsil6eufYp1z69LdCQRkSYqFj1I1pAsbnvlNgafN5inr3+at375VqIjiYgAKhY9TnpeOrcsuIXhlw/nuU88x/bfbU90JBERFYueqH9Wf276w02MmT6GTd/fxF///a+NV7KLiCSEikUPlZKWwo3P3Migqwbx8tdf5qWvvaSCISIJ057bqkqCJPdLZuy9Yyk5s4TXv/M6hw8c5rofX0dSsmq8iHQvFYsezpKMa394LWkD03j131/lSN0R/vkX/0xyaq++F5SI9DIqFr2AmXHlg1eSlpPGS197iaMHj3Ljb2+k34B+iY4mIn2E9mf0Ipd89RI+/NMPs/HFjfxq2q84Unck0ZFEpI9QsehlJs6ayPVPXc+217fx8yt+zqE9hxIdSUT6ABWLXujsmWfz0ec+SlJyEpZkiY4jIn2Ajln0Umdedyajp41WsRCRbqEti15MhUJEuouKhSTMH//4R8aMGcOoUaN46KGHTln+8MMPM27cOM455xyuvPJKtmzZ0rQsOTmZCRMmMGHCBKZPn35KXxHpWtoNJQlx/Phx7rrrLhYsWEBxcTEXXHAB06dPZ9y4cU1tzjvvPCorKxkwYAA//vGP+epXv8pvfvMbANLT01m+fHmi4ov0Odqy6GuqquCzn4XsbEhKCv/87GfD87vRG2+8wahRoygtLSU1NZWZM2fy/PPPn9RmypQpDBgwAICLLrqImhrdGEokUVQs+pIXX4RzzoHHH4eDB8E9/PPxx8PzX+y+W51v376dYcOGNU0XFxezfXvrI+w+8cQTTJs2rWn68OHDlJeXc9FFF/Hcc88FmlVEAt4NZWZTge8BycDj7v5Qs+X/AtwBNAC7gdvdfUtk2XFgZaTpVnfXjunOqKqCG26AQy1cl3HsWPhxww2wYgWMHBl4nJYGRTRr+YD9k08+SWVlJa+88krTvK1btzJ06FCqq6u54oorGD9+PCO7IbdIXxXYloWZJQOPAtOAccDHzGxcs2bLgHJ3Pwd4Bvh21LL33X1C5KFC0Vnf/W64ILTl2DF45JFuiVNcXMy2bduapmtqahg6dOgp7V566SUefPBBKioq6N+/f9P8xralpaVMnjyZZcuWBR9apA8LcjfUJGCTu1e7+1FgLjAjuoG7L3T3xv/qLgaKA8zTtz35ZPuKxS9/2S1xLrjgAjZu3Mjbb7/N0aNHmTt37ilnNS1btoxPf/rTVFRUMGjQoKb5+/fv58iR8FAne/bs4bXXXjvpwLiIdD0L6h4JZnYDMNXd74hM3wJc6O6zW2n/Q2Cnu38rMt0ALCe8i+ohdz9lx7SZzQJmARQWFk6cO3duh/PW19eTmZnZ4f5B6apcl19xBdaO37Wb8crLL3dLrsWLF/Poo49y4sQJpk2bxs0338ycOXMYM2YMl1xyCV/60pd4++23ycvLA6CwsJAHH3yQVatW8fDDD2NmuDvXX3891113XZflCoJyxUe54tOZXFOmTFnq7uUxG7p7IA/gRsLHKRqnbwF+0ErbmwlvWfSPmjc08rMU2AyMbOv9Jk6c6J2xcOHCTvUPSpflyspyDx/SbvuRnd29ubqYcsVHueJzOuYCKr0d3+lB7oaqAYZFTRcDO5o3MrOrgK8D0929aRhVd98R+VkNLALOCzDr6e/mm6FfjCHN+/WDW27pnjwi0qsEWSyWAKPNbISZpQIzgYroBmZ2HvBTwoViV9T8XDPrH3keAi4B1gSY9fT3pS+1r1jcfXf35BGRXiWwYuHuDcBsYD6wFnja3Veb2QNm1ngk8ztAJvBbM1tuZo3FpAyoNLO3gIWEj1moWHTGyJHwzDMwYMCpRaNfv/D8Z57pltNmRaT3CfQ6C3efB8xrNu++qOdXtdLvdWB8kNn6pGnTwtdRPPJI+Kyn+nrIzAzverr7bhUKEWmVxobqa0aOhB/+MPwQEWknDfchcVv0b4s4sPxAi1dhi8jpSVsWEpdDew/xxg/e4P297/POnHeYNHsS4z8+ntSM1ERHE5EAactC4jIgfwB3b7ubM79yJknJSbzw6Rd4pPgR/vTlP7G/en+i44lIQFQsJG790vsx5NohzHpzFp/86ycZefVIFv/3Yr4/6vv8evqvqVpQpV1UIqcZ7YaSDjMzSi4toeTSEuq211H5k0qW/nQpG36/gdDYEBfMvoBzP3Eu/bP6x34xEenRtGUhXSK7KJsrvnkFd2+7m4/84iOkZqXy4uwXebjoYV78wovs3bA30RFFpBO0ZSFdKqV/Cufeci7n3nIuNX+v4Y0fvEHljyt54/tvMGrqKCZ9bhKjpo7Cklq+d4WI9EwqFhKY4guLKb6wmKv/62qWPraUyp9U8tR1T5FTksP4j49n/MfHM+isQbFfSEQSTsVCApc5OJPL77ucS++5lLW/W8tbP3uL1779Gq/+x6sUnlvIOTefw9kfO5vsouxERxWRVqhYSLdJTk3m7I+ezdkfPZv6d+tZ/fRqVj65kgVfWcCCry5g+OThnHPzOZRdX0ZaTlqi44pIFBULSYjMwkwu/NyFXPi5C9m7cS8rf7WSlb9aScWnKvjDZ//AmH8aw/iPj2fUtFGk9NfHVCTR9FcoCZc/Op/J90/m8m9czo4lO1jx5ApWzV3FmmfWkJabxrgbx3HOx8+h5NISHRgXSRAVC+kxzIyiSUUUTSrimoevofqlalY8uYKVT67kzcfeJKckh7M+ehZjPzKWoguLSErWmd8i3UXFQnqkpJQkRk0dxaipozj63lHWP7+eFU+uYPEji3n9O6+TMSiD0R8ezdgZYym9qpR+A2Lc2ElEOkXFQnq81IxUxt80nvE3jefwgcNsfHEjGyo2sPaZtSyfs5yUtBRKP1TKmBljOJp7NNFxRU5LKhbSq6QNTGP8x8Yz/mPjOX70OFv+soX1FetZ//x6Nvx+Axhsu3AbY2aMYcz0MYTKQpjpOIdIZ6lYSK+VnJpM6VWllF5VytTvTeXdFe8y/3vzObLiCH++98/8+d4/kzcqjzOnn8mY6WMouaSEpBQd5xDpCBULOS2YGYPPHcwZnziDyZMnU1dTx4YXNrD++fUs+eESFj+8mLTcNEqvLKX0Q+ECk1uam+jYIr1GoMXCzKYC3wOSgcfd/aFmy/8FuANoAHYDt7v7lsiyW4F/jTT9lrv/PMiscnrJLs6m/M5yyu8s58jBI1T9qYqNL2ykakEVa55ZA0BuaS4jrhpB6VWljLhiBAPyByQ4tUjPFVixMLNk4FHgQ0ANsMTMKtx9TVSzZUC5ux8ys88A3wY+amZ5wDeAcsCBpZG+uruOxK1/Vn/GXT+OcdePw93Zu34v1S9VU72gmlW/XsWbj70JBkPOH9K01VFySQkpadrwFmkU5F/DJGCTu1cDmNlcYAbQVCzcfWFU+8XAzZHn1wAL3H1fpO8CYCrw6wDzSh9gZoTGhgiNDTFp9iRONJxg+5LtVC+opvqlav72X3/jtYdeIyUthZIPljQdExk8YbAuCJQ+LchiUQRsi5quAS5so/2ngBfb6FvUpelECF/PMeziYQy7eBiX33c5Rw4eYctftjQVj5e+9hIA6fnpnHHZGeGbPX2whMETBpPcLznB6UW6jwV1+0szuxG4xt3viEzfAkxy98+10PZmYDZwubsfMbOvAP3d/VuR5f8POOTu323WbxYwC6CwsHDi3LlzO5y3vr6ezMzMDvcPinLFp6tzHdlzhANvHmD/0v3Urqrl8I7DACSlJZE9Lpuc8TnkjM8he1w2yemtF4++sr66inLFpzO5pkyZstTdy2O1C3LLogYYFjVdDOxo3sjMrgK+TqRQRPWd3KzvouZ93f0x4DGA8vJynzx5cvMm7bZo0SI60z8oyhWfQHLd8I+nddvr2Prq1vDjr1vZ8ost4GDJxpDzhzRteZRcWkJGQUawubqAcsWnL+cKslgsAUab2QhgOzATuCm6gZmdB/wUmOruu6IWzQf+3cwaz228Grg3wKwi7ZJdlN00zDrA4drDbHt9W1PxWPKjJSx+ZDEA+WPymwrHoeRDuLsuEJReK7Bi4e4NZjab8Bd/MjDH3Veb2QNApbtXAN8BMoHfRv6Itrr7dHffZ2bfJFxwAB5oPNgt0pOk5aQxetpoRk8bDUDDkQZ2VO5g66tb2fbqNtY+s5Zljy8DYOXnV1J0QRFDJw1tGjAxs7Dn7dIQaUmg5wa6+zxgXrN590U9v6qNvnOAOcGlE+l6Kf1TKLmkhJJLSuBr4CecXat38fIvXiazNpPtb2zn1f94FT8ePlaYc0ZOU+EomlTEkPOHkJqZmuB/hcipdCK5SIAsySgcX8iQ64Y07VM++t5Rdi7byfY3tjc91vx2TVP7grMKwsXjwnABGXTWIA1TIgmnYiHSzVIzUsMHwi8taZr33u732LFkBzV/r2HHGztY97t1LHsivPsqJS2FQeMHMfi8wQw5bwiDzxtM4fhCDcsu3UrFQqQHyCjIYPS1oxl9bfjYh7uzv3o/2/++nXfefIedy3ay5rdrwlebE94CyR+T31Q8GgtJel56Iv8ZchpTsRDpgcyMvJF55I3MY/xN44FwAandWsvOZTvZuXwnO5ftZMtft7DyqZVN/XJKchg8YXBTARk8YTA5JTk6C0s6TcVCpJcwMwaeMZCBZwxk7EfGNs0/tOcQO5fv5J1l4S2Qnct2sv7368OjqgH9s/sz6OxBFJxdQOH4QgadPYhB4wdp4ESJi4qFSC83IDSgaQyrRkffO8qulbvYuXwnu1btYteqXax9Zm3TbiyAzMFalAV1AAAL2klEQVSZpBSlcOSyI00FpGBcAakZOhtLTqViIXIaSs1IpfiiYoovKm6a5+7U76xn18pdTQWk6vUqKn9SScP7DeFGFh66fdDZg8JbI+MKCI0NkT8mX0Wkj1OxEOkjzIysIVlkDcli5NUjgfAwEZd98DIOvH2AXat28e7Kd9m9aje7Vu1iwwsbmq4HgfA1IaGxIUJlIQrKCgiVhUfvjR7WRE5fKhYifVxSchJ5o/LIG5V30rGQhiMN7K/az+61u9mzdk/4sW4Pb/71TY4dOtbULj0/nYKyAvLH5p9URAaeMVDDup9GVCxEpEUp/VMoGFdAwbiCk+b7Cad2Wy171oULSGMxWf/8+qahTQCS+yeTNyqP/NH55J0Z/pl/Zj55o/PIHJypM7R6GRULEYmLJf3jrKxR14w6admhvYeaCsi+jfvYu2EvezfsZeO8jRw/erypXWpmKnmjWy4kOkurZ1KxEJEuMyB/wClXpwOcOH6Cum114eKxMVxA9m3cx46lO1jz7JqTjo2k56WTMiiFveftJbc0l9yRueSW5pI3Mo+soVnatZUgKhYiErik5CQGDh/IwOEDmw6uNzp+9Dj7395/0pZIVWUVNYtrWP306pMKSXL/ZHJHRApIVBHJHZnLwOED6ZeuIVCComIhIgmVnJpMaEyI0JhQ07zGm/kcP3ac2q217K/az/7q/eyr2tf0fMsrWzhaf/Sk18oqyiK3NFw4Bo4Y2FSgckfkkl2crQEZO0HFQkR6rOR+yU3DnjTn7hzafeiUIrK/ej+bF22m7sm6pqvYIXw3w+zi7KbikTM8p+n5wOEDySrKIilZxaQ1KhYi0iuZGRmDMsgYlHHSxYeNjh89Tu22Wg5sPvCPx9vhn1ULqji44+BJxSQpJYnsYdkMPGMgOSU5ZJdkk1OSc9KjL1OxEJHTUnJq61slEL6OpHbrqcWkbltdeMtke91Jx0sAUrJTWF+6npwzck4pJDklOWQUZpy2WycqFiLSJ6X0Twmfsjs6v8XlJxpOcPCdg9RuraV2Sy21W2tZ+7e1ZBzPaNrVdaT2yEl9klKSyBqaRXZxNtnF2WQVZ5EzLKdpOrs4m8zBmb3y2ImKhYhIC5JSksgZlkPOsBy4JDyvYVFD0x0PAQ7XHqZuW114C2XLAepq6jhYc5C6mjp2Lg+P/ts07laEJYeHXYkuKNnF2WQXZZNVlEXW0PCjp53ZpWIhItJBaTlppOWkMejsQS0ud3cO7z9MXU0ddTV11G6rPamgvLvyXTbO23jS8ClNr52bFi4gkeLRVEgiP7OLsskYlNFtWymBFgszmwp8D0gGHnf3h5otvwz4b+AcYKa7PxO17DjQeFeXre4+PcisIiJdzcxIz0snPS+dwnMKW2zj7hw+cJiDOw6GH9vDP+u211G/o5667XXsWr2L+p31pxxDsSQjozCD9LHpJ23xBCGwYmFmycCjwIeAGmCJmVW4+5qoZluB24Avt/AS77v7hKDyiYj0BGZGem466bnpDDqr5S0UCF8F/96u904pKAd3HGTfkX2B5wxyy2ISsMndqwHMbC4wA2gqFu6+ObLsRIA5RER6vaTkpKYh5pl48rJFixYF/v7m7rFbdeSFzW4Aprr7HZHpW4AL3X12C21/BrzQbDdUA7AcaAAecvfnWug3C5gFUFhYOHHu3LkdzltfX09mZmaH+wdFueKjXPFRrvicjrmmTJmy1N3LYzZ090AewI2Ej1M0Tt8C/KCVtj8Dbmg2b2jkZymwGRjZ1vtNnDjRO2PhwoWd6h8U5YqPcsVHueJzOuYCKr0d3+lBHkavAYZFTRcDO9rb2d13RH5WA4uA87oynIiItF+QxWIJMNrMRphZKjATqGhPRzPLNbP+kechwmc5r2m7l4iIBCWwYuHuDcBsYD6wFnja3Veb2QNmNh3AzC4wsxrCu6x+amarI93LgEozewtYSPiYhYqFiEiCBHqdhbvPA+Y1m3df1PMlhHdPNe/3OjA+yGwiItJ+vW+AEhER6XYqFiIiElNg11l0NzPbDWzpxEuEgD1dFKcrKVd8lCs+yhWf0zHXGe5eEKvRaVMsOsvMKr09F6Z0M+WKj3LFR7ni05dzaTeUiIjEpGIhIiIxqVj8w2OJDtAK5YqPcsVHueLTZ3PpmIWIiMSkLQsREYmpTxULM5tqZuvNbJOZ3dPC8v5m9pvI8r+b2fBuyDTMzBaa2VozW21mX2ihzWQzqzWz5ZHHfS29VkD5NpvZysj7Vraw3Mzs+5F1tsLMzu+GTGOi1sVyM6szsy82a9Mt68zM5pjZLjNbFTUvz8wWmNnGyM/cVvreGmmz0cxu7YZc3zGzdZHf0+/MbGArfdv8nQeQ634z2x71u7q2lb5t/v0GkOs3UZk2m9nyVvoGub5a/H5IyGesPUPTng4Pwrd2rSI85Hkq8BYwrlmbzwI/iTyfCfymG3INAc6PPM8CNrSQazLh+30kYr1tBkJtLL8WeBEw4CLg7wn4ve4kfK54t68z4DLgfGBV1LxvA/dEnt8D/GcL/fKA6sjP3Mjz3IBzXQ2kRJ7/Z0u52vM7DyDX/cCX2/F7bvPvt6tzNVv+XeC+BKyvFr8fEvEZ60tbFk137nP3o0DjnfuizQB+Hnn+DHClmVmQodz9HXd/M/L8IOFBF4uCfM8uNgP4hYctBgaa2ZBufP8rgSp378wFmR3m7n8Bmt/TMvpz9HPgIy10vQZY4O773H0/sACYGmQud/+Thwf4BFhMC+OyBa2V9dUe7fn7DSRX5Dvg/wK/7qr3a682vh+6/TPWl4pFEbAtarqGU7+Um9pE/qhqgfxuSQdEdnudB/y9hcUXm9lbZvaimZ3VXZkAB/5kZkstfGfC5tqzXoM0k9b/iBO1zgrd/R0I/7EDLd1YOdHr7XbCW4QtifU7D8LsyO6xOa3sUknk+vog8K67b2xlebesr2bfD93+GetLxaKlLYTmp4K1p00gzCwTeBb4orvXNVv8JuHdLOcCPwBOucVsgC5x9/OBacBdZnZZs+WJXGepwHTgty0sTuQ6a49ErrevE75d8a9aaRLrd97VfgyMBCYA7xDe5dNcwtYX8DHa3qoIfH3F+H5otVsL8zq8zvpSsWjPnfua2phZCpBDxzaZ42Jm/Qh/EH7l7v/bfLm717l7feT5PKCfhW8KFTj/xx0LdwG/I7w7IFqn7ojYSdOAN9393eYLErnOgHcbd8VFfu5qoU1C1lvkIOeHgY97ZMd2c+34nXcpd3/X3Y+7+wngf1p5v0StrxTg/wC/aa1N0Ourle+Hbv+M9aVi0Z4791UAjWcM3AC83NofVFeJ7A99Aljr7g+30mZw47ETM5tE+Pe2N8hckffKMLOsxueED5CuatasAviEhV0E1DZuHneDVv/Hl6h1FhH9OboVeL6FNvOBqy18V8hcwut2fpChzGwq8DVgursfaqVNe37nXZ0r+hjXP7fyfh2+82YnXQWsc/ealhYGvb7a+H7o/s9YEEfwe+qD8Jk7GwifVfH1yLwHCP/xAKQR3qWxCXgDKO2GTJcS3jRcASyPPK4F7gTujLSZDawmfAbIYuAD3bS+SiPv+Vbk/RvXWXQ2Ax6NrNOVQHk3ZRtA+Ms/J2pet68zwsXqHeAY4f/JfYrwca4/AxsjP/MibcuBx6P63h75rG0CPtkNuTYR3ofd+DlrPPNvKDCvrd95wLl+GfnsrCD8JTikea7I9Cl/v0Hmisz/WeNnKqptd66v1r4fuv0zpiu4RUQkpr60G0pERDpIxUJERGJSsRARkZhULEREJCYVCxERiUnFQkREYlKxEBGRmFQsRAJiZhdEBsdLi1zpu9rMzk50LpGO0EV5IgEys28RHhkgHahx9/9IcCSRDlGxEAlQZByjJcBhwkOOHE9wJJEO0W4okWDlAZmE73KWluAsIh2mLQuRAJlZBeG7uo0gPEDe7ARHEumQlEQHEDldmdkngAZ3f8rMkoHXzewKd3850dlE4qUtCxERiUnHLEREJCYVCxERiUnFQkREYlKxEBGRmFQsREQkJhULERGJScVCRERiUrEQEZGY/j90CFcMaUDj7QAAAABJRU5ErkJggg==\n",
630            "text/plain": "<matplotlib.figure.Figure at 0x7fb64b566fd0>"
631          },
632          "metadata": {}
633        },
634        {
635          "output_type": "stream",
636          "text": "Limit as x -> 4 = 0.25\n",
637          "name": "stdout"
638        }
639      ]
640    },
641    {
642      "metadata": {},
643      "cell_type": "markdown",
644      "source": "## Rules for Limit Operations\nWhen you are working with functions and limits, you may want to combine limits using arithmetic operations. There are some intuitive rules for doing this.\n\nLet's define two simple functions, ***j***:\n\n\\begin{equation}j(x) = 2x - 2\\end{equation}\n\nand ***l***:\n\n\\begin{equation}l(x) = -2x + 4\\end{equation}\n\n\nRun the cell below to plot these functions:"
645    },
646    {
647      "metadata": {
648        "trusted": true
649      },
650      "cell_type": "code",
651      "source": "%matplotlib inline\n\n# Define function j\ndef j(x):\n    return x * 2 - 2\n\n# Define function l\ndef l(x):\n    return -x * 2 + 4\n\n\n# Plot output from functions j and l\nfrom matplotlib import pyplot as plt\n\n# Create an array of x values\nx = range(-10, 11)\n\n# Get the corresponding y values from the functions\njy = [j(i) for i in x]\nly = [l(i) for i in x]\n\n# Set up the graph\nplt.xlabel('x')\nplt.xticks(range(-10,11, 1))\nplt.ylabel('y')\nplt.yticks(range(-30,30, 2))\nplt.grid()\n\n# Plot x against j(x)\nplt.plot(x,jy, color='green', label='j(x)')\n\n# Plot x against l(x)\nplt.plot(x,ly, color='magenta', label='l(x)')\n\nplt.legend()\n\nplt.show()\n",
652      "execution_count": 26,
653      "outputs": [
654        {
655          "output_type": "display_data",
656          "data": {
657            "image/png": 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\n",
658            "text/plain": "<matplotlib.figure.Figure at 0x7fb64b59af98>"
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666      "cell_type": "markdown",
667      "source": "### Addition of Limits\n\nFirst, let's look at the rule for addition:\n\n\\begin{equation}\\lim_{x \\to a} (j(x) + l(x)) = \\lim_{x \\to a} j(x) + \\lim_{x \\to a} l(x)\\end{equation}\n\nWhat we're saying here, is that the limit of *j(x)* + *l(x)* as *x* approaches *a*, is the same as the limit of *j(x)* as *x* approaches *a* added to the limit of *l(x)* as *x* approaches *a*.\n\nLooking at the graph for our functions ***j*** and ***l***, let's apply this rule to an *a* value of **8**.\n\nBy visually inspecting the graph, you can see that as *x* approaches 8 from either direction, *j(x)* gets closer to 14, so:\n\n\\begin{equation}\\lim_{x \\to 8} j(x) = 14\\end{equation}\n\nSimilarly, as *x* approaches 8 from either direction, *l(x)* gets closer to -12, so:\n\n\\begin{equation}\\lim_{x \\to 8} l(x) = -12\\end{equation}\n\nSo based on the addition rule:\n\n\\begin{equation}\\lim_{x \\to 8} (j(x) + l(x)) = 14 + -12 = 2\\end{equation}\n\n### Subtraction of Limits\nHere's the rule for subtraction:\n\n\\begin{equation}\\lim_{x \\to a} (j(x) - l(x)) = \\lim_{x \\to a} j(x) - \\lim_{x \\to a} l(x)\\end{equation}\n\nAs you've probably noticed, this is consistent with the rule of addition. Based on an *a* value of 8 (and the limits we identified for this *a* value above), we can apply this rule like this:\n\n\\begin{equation}\\lim_{x \\to 8} (j(x) - l(x)) = 14 - -12 = 26\\end{equation}\n\n### Multiplication of Limits\nHere's the rule for multiplication:\n\n\\begin{equation}\\lim_{x \\to a} (j(x) \\cdot l(x)) = \\lim_{x \\to a} j(x) \\cdot \\lim_{x \\to a} l(x)\\end{equation}\n\nAgain, you can apply this to the limits as x approached an *a* value of 8 we identified previously:\n\n\\begin{equation}\\lim_{x \\to 8} (j(x) \\cdot l(x)) = 14 \\cdot -12 = -168\\end{equation}\n\nThis rule also applies to multipying a limit by a constant:\n\n\\begin{equation}\\lim_{x \\to a} c \\cdot l(x) = c \\cdot \\lim_{x \\to a} l(x)\\end{equation}\n\nSo for an *a* value of 8 and a constant *c* value of 3, this equates to:\n\n\\begin{equation}\\lim_{x \\to 8} 3 \\cdot l(x) = 3 \\cdot -12 = -36\\end{equation}\n\n\n### Division of Limits\nFor division, assuming the limit of *l(x)* when x is approaching *a* is not 0:\n\n\\begin{equation}\\lim_{x \\to a} \\frac{j(x)}{l(x)} = \\frac{\\lim_{x \\to a} j(x)}{\\lim_{x \\to a} l(x)}\\end{equation}\n\nSo, based on our limits for *j(x)* and *l(x*) when *x* approaches 8:\n\n\\begin{equation}\\lim_{x \\to 8} \\frac{j(x)}{l(x)} = \\frac{14}{-12}= \\frac{7}{-6}\\end{equation}\n\n### Limit Exponentials and Roots\n\nAssuming *n* is an integer:\n\n\\begin{equation}\\lim_{x \\to a} (j(x))^{n} = \\Big(\\lim_{x \\to a} j(x)\\Big)^{n}\\end{equation}\n\nSo for example:\n\n\\begin{equation}\\lim_{x \\to 8} (j(x))^{2} = \\Big(\\lim_{x \\to 8} j(x)\\Big)^{2} = 14^{2} = 196\\end{equation}\n\nFor roots, again assuming *n* is an integer:\n\n\\begin{equation}\\lim_{x \\to a} \\sqrt[n]{j(x)} = \\sqrt[n]{\\lim_{x \\to a} j(x)}\\end{equation}\n\nSo:\n\n\\begin{equation}\\lim_{x \\to 8} \\sqrt[2]{j(x)} = \\sqrt[2]{\\lim_{x \\to 8} j(x)} = \\sqrt[2]{14} \\approx 3.74\\end{equation}\n"
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