CHAP4_eqs.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 기본적인 import들\n",
    "# 이후 그래프 그리는 코드에는 중복으로 적지 않음.\n",
    "# 다른 곳으로 그래프 그리는 코드를 복사-붙이기 할 때는 \n",
    "# 이 import 코드와 함께 복사-붙이기 해야함\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib as mpl\n",
    "from mpl_toolkits import mplot3d\n",
    "import matplotlib.font_manager as mfm\n",
    "\n",
    "# 그래프에 한글 폰트 사용을 위한 설정\n",
    "# https://financedata.github.io/posts/matplotlib-hangul-for-ubuntu-linux.html\n",
    "path = '/usr/share/fonts/truetype/nanum/NanumBarunGothic.ttf'\n",
    "fontprop = mfm.FontProperties(fname=path, size=18)\n",
    "\n",
    "# numpy 출력 형식 지정\n",
    "np.set_printoptions(precision=4, linewidth=150)\n",
    "\n",
    "# matplotlib 스타일 지정\n",
    "mpl.style.use('bmh')\n",
    "mpl.style.use('seaborn-whitegrid')\n",
    "style = plt.style.library['bmh']\n",
    "# 스타일 컬러를 쉽게 쓸 수 있도록 리스트 저장\n",
    "style_colors = [ c['color'] for c in style['axes.prop_cycle'] ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 지수함수의 극한\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "지면 관계로 책에 다 싣지 못한 지수함수의 특별한 형태에 대한 극한을 설명합니다.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. $\\lim_{x \\to 0} \\dfrac{e^x - 1}{x} = 1$\n",
    "\n",
    "주어진 식은 $x=0$일 때 $\\frac{0}{0}$이 되므로 함수값이 정의되지 않습니다. 위 함수의 그래프를 그려서 $x=0$ 근처에서 어떤 형상인지 확인해보도록 합시다.  위 함수는 분자와 분모의 비를 함숫값으로 가지고 있습니다. 따라서 $x=0$ 근처에서 분모 $x$와 분자 $e^x - 1$이 함수로서 어떤 형태를 띄는지 비교해보는 것은 극한의 직관을 얻기 위한 좋은 방법입니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x504 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = lambda x : np.exp(x) - 1\n",
    "g = lambda x : x\n",
    "\n",
    "fig = plt.figure(figsize=(15,7))\n",
    "\n",
    "ax1 = fig.add_subplot(1, 2, 1)\n",
    "ax1.xaxis.set_tick_params(labelsize=18)\n",
    "ax1.yaxis.set_tick_params(labelsize=18)\n",
    "x = np.linspace(-0.5, 0.5, 100)\n",
    "ax1.plot(x, f(x), 'k', label=r\"$e^x - 1$\")\n",
    "ax1.plot(x, g(x), 'k--', label=r\"$x$\")\n",
    "ax1.legend(fontsize=25)\n",
    "ax1.set_xlabel(r'$x$', fontsize=25)\n",
    "ax1.set_ylabel(r'$y$', fontsize=25)\n",
    "\n",
    "ax2 = fig.add_subplot(1, 2, 2)\n",
    "ax2.xaxis.set_tick_params(labelsize=18)\n",
    "ax2.yaxis.set_tick_params(labelsize=18)\n",
    "x = np.linspace(-0.05, 0.05, 100)\n",
    "ax2.plot(x, f(x), 'k', label=r\"$e^x - 1$\")\n",
    "ax2.plot(x, g(x), 'k--', label=r\"$x$\")\n",
    "ax2.legend(fontsize=25)\n",
    "ax2.set_xlabel(r'$x$', fontsize=25)\n",
    "ax2.set_ylabel(r'$y$', fontsize=25)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "그래프를 확인해보면 $e^x-1$과 $x$는 0 근방에서 거의 똑같은 함수입니다. 따라서 $x=0$ 근방에서 두 함수의 비는 1이 될 것이라 예상할 수 있습니다. 직접 $\\dfrac{e^x-1}{x}$를 그려 보면 다음과 같습니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ISCulpqbCwcEBz549Q48ePXjX4XLCUiEirZObmwsXFxfcunULLVq0wN69e6Gvry92rApBV+wARB/ixYsXOHjwIHJyctCgQQPI5XI8evQIZmZmcHd3L/c82dnZOH/+PFavXo358+fD2tq63DNQ8RQKBcaMGYNTp06hXr16OHLkCGrUqCF2rAqDpUIa4ciRI5g2bRoEQUCbNm2wa9cu3L9/H48ePSrXHElJSZg5cyaMjIygp6eHW7duQS6Xl2sGere5c+di586dMDIywqFDh/Dxxx+LHalCYamQ5OXn58Pa2hqVKlXCvXv3YGZmhhYtWohyi/KPPvoImzdvBgDs3bsXhw4dKvcM9Ha//PILfH19oauriz179qBt27ZiR6pweEyFJE9PTw/m5uYAgEuXLvGiNSrWgQMHMGnSJADA+vXrYWdnJ3KiiomlQhpBEAQAwJ9//ol27doBANLS0nD//n0xY5FEXLx4Ea6urlAoFPDx8cGYMWPEjlRhsVRI8sLCwjBx4kRkZ2fj7Nmzyvs1HTx4ECYmJiKnI7Hdv38fTk5OyMrKwujRo+Hj4yN2pAqNx1RIdP/88w+2bt0KPT096OrqIj4+HpMmTUKTJk0AALVq1ULDhg2xbds2/PTTTzh27Bju3LmDFi1aoFq1aiKnJzElJyfDwcEBiYmJ6NOnD9avX89rUUTGUiFRHT16FL6+vli+fLlyWuvq1asIDg7GnDlzAADW1taoV68e6tWrBwDo2rWraHlJOrKzszFgwADcvn0bbdq0QUhICPT09MSOVeFx+otEc+PGDcycORMzZsxQFsrjx4+xfft29O7dW+R0JGUKhQIeHh44d+4cGjRogEOHDvG5KBLBPRUSzcqVK6Grq4vr16/j+vXrkMlkMDExwVdffYU6deqo5DOePn0KLy+vEl1LMm7cOAwZMkQln0+qJwgCZsyYgZCQEBgbG+Pw4cOoX7++2LHo/7BUSBQFBQU4d+4cevbsidmzZ6vtc+rXr89rSbTM0qVL4efnBz09PYSGhqJVq1ZiR6J/4fQXiSI1NRUFBQW82plKZMuWLZg9ezZkMhm2bt2Knj17ih2J3sBSIVHUqlULxsbGb52W2rJlS/kGIsk7cOAAxo8fD+DV1Kmrq6vIiag4LBUNl5CQgClTpmDYsGGws7PD77//Xux6jx49Qvfu3eHr61vOCYuno6MDNzc3nDlzBvn5+crlycnJWLhwIZo2bSpiug+TnJwM4NVeF6nXuXPnMHToUMjlcsybNw+TJ08WOxK9BY+paLC8vDx8+eWXWLBgAZo3b44vvvgCc+bMQcuWLZXXeLy2c+dOJCQk4M6dOyKlLWry5MmQyWSYOnUqGjVqBLlcjipVqmDy5MmSvqjR1dUVKSkpePjwIQBg+vTpWL16NZo0aYJVq1aJnE77XL9+HU5OTsjJyYGnpye+++47sSPRO7BUNNjmzZvh4OCA5s2bA3h1B125XI7Y2NgipRIVFQUA6NixY7Hv9c8//8Db27vQXkNZjB8/HiNGjHjnOjo6Opg6dapKPq88BQUFiR2hwnj48CHs7e3x4sULODs7Y+3atby4UeJYKhpKLpdj//79CA0NBfBqGuzvv/9GpUqV0L59+0LrpqWl4datWwCAzp07F/t+//nPf3Ds2DH1hiYqgaSkJNjb2yMuLg7dunVDYGAgdHX5I0vqeExFQz19+hTOzs7Kp9nt2bMHCoUCn376aZFrPC5evAiFQoGqVauiZcuWYsQlKpGMjAw4OjoiNjYWrVu3xv79+2FoaCh2LPoALBUN1bBhQ3h6eir/e8+ePQBQ7JTT66kva2tr/qZHkpeXl4dBgwbh4sWLaNy4MY4ePconN2oQlooWuHLlCp48eYK6devC1ta2yOuvS+WTTz4p72hEJfL6UcBhYWEwMTFBWFgY6tatK3YsKgFJl0pMTAzs7e1haWmJe/fuffB2f/zxB7p27QpLS0vk5uaqMaE0HD16FADQu3dvVKpU+Ct98eIFYmNjAbBUSNoEQcCiRYsQGBgIIyMjHDlyRPlwNtIckp0LCQwMxJIlS0p0kzi5XI41a9YgICCgQu0uX7lyBUDxpREdHQ1BEFC9enU0a9bsre9R3md/WVpaquRzpOx1mdOH8fX1xcaNG5W3X+ETPjWTJEslOjoaS5YswcKFCxEfHw9/f/8P2m7//v3Yvn071qxZg6NHjyrPjNJ2ry++K+6WJ6+nvjp16lRkL+bfyvvsr5L+wI2Li1Pe+p60z6+//oo5c+ZAJpPxLtUaTpLTXzVq1EBQUBAGDx5cou0aNmyIPXv24NNPP1VPMIkyMzMD8Go++t8ePHigvMKeU18kVfv371eedLJ48WIMHTpU5ERUFpLcU7GwsCjVdhV1d9nNzQ1RUVEICQnBggULALx6lvvXX3+N9PR0AG+/PkUb3bt3Dzt37kTlypURGxuLKVOmIDU1FWfPnoVCoUDdunWV95AicZ05cwbDhg2DQqHAggULMHr0aLEjURlJck+FSsbe3h5+fn64cuUKBgwYgEGDBmHnzp3K3/hq165dYQ54xsfHIyAgALNmzcLMmTNhZ2eHSZMm4a+//sK8efNw4cIFrF27VuyYBODatWvo378/cnJy4OXlhUWLFokdiVRAknsqVHKOjo5wdHQstGz69OkAgF69eokRSRTr16/HzJkzlReFZmdnIzExEe7u7gCAAQMGoFGjRmJGJLyamrW3t8fLly/h4uLC269oEZYKXh0E1iTp6emIi4tDZmYmfv75Z6SlpWHRokWoWrWqcp2XL18iLCwMMpkMPXv21Lgxvun1mN+nX79+yM/PV6577tw5NGnSBHl5eYiLi4OTkxMAzfjOP3TMmub58+dwcXFBfHw8bGxssGzZMiQkJADQ3jG/i7aNmaUCaNxZRa/PhAoICMCJEycAvHqS4r/HsWXLFuTn52PYsGHo0aOHWFFV5kPP/vr3OgqFAlevXsXAgQM17jsGtPOMt5SUFIwcORL//PMP2rZtiyNHjqB69erK17VxzO+jqWOOj48vdjmPqWiwhIQE6OrqwsXFBVZWVsrlUVFRCAwMhLW1NebOnStiQnFdv34dL168gI2NjXJZTk6O8rdiKl/p6elwdHTEtWvXYGlpiWPHjhUqFNIOLBUN1qtXL1haWmLWrFnKZadOncLkyZNhb2+PgIAAGBgYiJiwfOXl5eHnn3/Gn3/+CeDVmUUACt21OTg4GM+fPxclX0WWk5MDZ2dnREVFoVGjRggPD4epqanYsUgNJDn99eTJE+UFfYmJiQCAu3fvIisrC8Crq7EPHz6MuXPnYv369ejatSsA4P79+8jMzATw6vYkAHDjxg3o6ekBAFq1alWu41A3W1tbpKWlYeLEicqDnCYmJli5cmWh384ripMnT2L9+vWoUaMGzM3NcfHiRejo6ChvohkbG4tHjx7Bw8ND5KQVS35+PoYOHYrIyEjUqVMH4eHhaNCggdixSE0kWSr+/v5FroafMmWK8s8RERFQKBSQy+WFLvjz8fFBdHR0oe2GDx+u/LM23jbDwcEBDg4OYseQhI4dO8LZ2RnJyclYvnw5Vq1ahV27dmHmzJlo0KABjIyMCu3VkfrJ5XKMHj0aBw4cQK1atXD8+HGNeFQ0lZ5MEARB7BBiunTpksZdNKmpB/bKgmPWPIIgwNvbG+vXr4eRkREiIyPf+uTR1zR9zKWhqWN+289OHlMhIpUTBAGzZ8/G+vXrYWBggAMHDry3UEg7sFSISOV++OEHLF26FLq6uggJCalw9+OryFgqRKRSq1atwvz585V3HO7Xr5/YkagcsVSISGW2bNmCqVOnAnh1y5xhw4aJnIjKG0uFiFRiz549GDduHABg+fLlvBN0BcVSIaIyO3bsGIYPHw6FQgEfHx9MmzZN7EgkEpYKEZXJmTNn4OLigvz8fHz55Zfw8fEROxKJiKVCRKV2+fJlODk5ITs7G+PGjcPy5ct5C/sKjqVCRKVy8+ZN2NnZIS0tDUOHDkVAQAALhVgqRFRy//zzD/r06YPk5GQ4ODhg27Zt0NHRETsWSQBLhYhK5PHjx+jVqxfi4uLQvXt3hISEKJ+0ScRSIaIPFhcXh549e+Kff/5Bx44dceDAAVSpUkXsWCQhLBUi+iAJCQno2bMn7t69i3bt2uHYsWMwNjYWOxZJDEuFiN7r+fPn6NWrF2JjY9GqVSscP34cNWvWFDsWSRBLhYjeKTk5Gb1798aNGzfQvHlzhIeHo3bt2mLHIoliqRDRW6WmpsLOzg5///03LCwsEBERwccA0zuxVIioWC9fvoS9vT0uX76MJk2aKB8HTPQuLBUiKiI9PR2Ojo64ePEiGjdujMjISNSvX1/sWKQBWCpEVEhmZiacnJxw/vx5fPzxxzhx4gQaNmwodizSECwVIlLKzs7GZ599htOnT6NevXo4ceIEGjduLHYs0iAlLpW4uDhcu3YNCoVCHXmISCQ5OTlwdnZWHjuJjIxEkyZNxI5FGka3pBsMGTIEKSkpqFKlCjp06IDOnTujc+fOaNGiBW8mR6Sh8vLyMHjwYISFhcHExAQRERGwtLQUOxZpoBKXyo8//ojg4GBcunQJp0+fxunTpyGTyWBkZIQOHTrgk08+QefOnWFlZaWOvESkYvn5+Rg2bBgOHTqE2rVrIyIiAs2bNxc7FmmoEpdK9+7d0b17dwDAnTt3EB0djejoaFy6dAknT57EqVOnAADGxsbo2bMnRo4cyf+DEklUQUEB3NzcsG/fPtSoUQPHjx9Hq1atxI5FGqzEpfJv5ubmMDc3h5ubGwDg3r17OHjwIDZs2IDc3FyEhoZi//798PLywtSpU1USmIhUQy6XY9SoUdi9ezeMjY0RFhaGdu3aiR2LNFyZSuVNTZo0wdSpU1G3bl28fPkS9evXx549e/DLL78gJycHs2fPVuXHEVEpKRQKjBs3DoGBgTAyMsLRo0fRsWNHsWORFijVKcW5ubnIy8t76+tDhw7F6dOn4ejoiE2bNmHNmjUICQlBVFRUqYMSkWooFApMmDABv/32G6pUqYLDhw/DxsZG7FikJUpcKqGhobC2tkaHDh0wceJEREREoKCgoMh6mZmZyj/36tULy5cvx+bNm8uWlojKRKFQwNPTExs3boSBgQEOHjyIbt26iR2LtEiJp7/WrVuHkSNH4t69ezh9+jROnDgBY2Nj2NraomXLljAyMsKFCxdQqVLhvurWrRv8/f1VFpyISkYul2PcuHH47bffYGhoiIMHD6JHjx5ixyItU+JSqV69OmbNmgUASElJwd69e3HgwAEcPnwYhw8fBgAYGBhg3bp1AIDffvsNpqamcHBwgJGRkQqjE9GHksvlGDNmDLZt24YqVarg0KFD+PTTT8WORVqoxKXi4uKCb7/9Fl9//TVq1aqF8ePHY/z48Xj69Cnu3r2L7OxsdOjQASYmJgCAtWvXIiMjAwkJCWjQoIHKB0BE71ZQUIBRo0YhMDAQVatWxeHDh5WXBRCpWolLZfjw4Th37hw+//xzWFlZYcaMGQCA+vXrF3sX06VLlyIwMBBxcXGYM2dO2RMT0QcrKCjAyJEjERQUBCMjIxw5cgRdu3YVOxZpsVKdUmxrawtbW1vcu3fvvev++2JJIio/+fn5cHNzw+7du1GtWjUcPXoUXbp0ETsWabkyXafCm80RSVN+fj6GDx+OPXv2wNjYGMeOHcMnn3widiyqAFR68SMRiS8vLw/Dhg3Dvn37UL16dYSFhaFTp05ix6IKgqVCpEVyc3MxdOhQ7N+/X3kvL2tra7FjUQXCUiHSErm5uRg0aBAOHTqEmjVrIjw8HO3btxc7FlUwLBUiLZCTk4OBAwfiyJEjqFWrFsLDw3lzSBIFS4VIw2VnZ8PFxQXHjh1TPg+lTZs2YseiCoqlQqTBsrKy4OzsjOPHjyuf2MjnoZCYWCpEGiorKwv9+/dHZGQkTE1NERkZiRYtWogdiyo4lgqRBsrMzISTkxNOnjwJMzMzREZG8gmrJAmlep4KEYknPT0djo6OOHnyJOrWrYuTJ0+yUEgyuKdCpEFSU1PRt29fREdHo169ejhx4gQsLCzEjkWkJOk9lZiYGNjb28PS0vKD7jMGANHR0XB3d0fbtm3RvrhjdPwAACAASURBVH17eHl5ISYmRs1JidQvMTERPXr0QHR0NBo3bowzZ86wUEhyJFsqgYGBGDp0aKEnSL7PpUuXMHbsWBgaGmLNmjX48ccfkZaWBnd3dzx58kSNaYlURxAE3LhxA1FRUbhx4wYEQcDTp0/RvXt3XL16FRYWFjhz5gz++9//ih2VqAhJTn9FR0djyZIlWLhwIeLj4z/4iZErVqzARx99hDVr1kBfXx8A0LFjR/Ts2RO//PILvv/+e3XGJioTQRCwadMmrFixAjdu3FAut7CwQEpKCpKSktC6dWuEhYXBzMxMxKREbyfJPZUaNWogKCgIgwcP/uBtXrx4gYsXL6JPnz7KQgGAWrVqwdbWFuHh4eqISqQSgiBgwoQJ8PT0hIGBAQICAnD8+HEEBASgatWqSEpKgomJCSIjI1koJGmSLBULC4sSn81y+/ZtCIJQ7Bxz06ZN8eLFC8THx6sqIpFKbdq0CRs2bMCcOXNw8eJFeHl5oXfv3vDy8sKlS5fw9ddf4/nz59i3b5/YUYneSZKlUhrJyckAgJo1axZ57fWy1+sQSYkgCFixYgU6dOiA77//HjKZrNDrMpkMP/zwA9q3b48VK1ZAEASRkhK9n9aUSm5uLgBAT0+vyGuvl+Xk5JRrJqIPcfPmTdy4cQNeXl5FCuU1mUwGLy8vXL9+Hbdu3SrnhEQfTpIH6kvDwMAAwKsn3r3p9TJDQ8Nit42Li1NfMDVIT0/XuMxlpc1jvn37NgC892yu16/HxsaiRo0aas8lBm3+nt9G28asNaXy0UcfAXh1cdibkpKSAAAmJibFbluvXj31BVODuLg4jctcVto85tfHAe/fv//O9V6/bmlpqbV/F9r8Pb+Npo75bceotWb6y9LSEjo6OoiNjS3yWmxsLExMTGBqaipCMqJ3a968OT7++GOsW7furcdLBEHA+vXr0bJlS1hZWZVzQqIPpzWlUq1aNXTp0gVHjx4tdOwkISEBf/zxBxwcHERMR/R2mzZtwuPHj/HXX39h3rx5RYpFEATMnTsXly9fxpdffvnW4y5EUiDJUnny5AmuXbuGa9euITExEQBw9+5d5bK8vDzs27cPzZs3x9mzZ5XbffXVV0hPT8cXX3yB8+fP48SJE/D29kb16tUxYcIEsYZD9FYrV66Ep6cnAMDa2ho//vgjrK2tC12nYm1tjSVLlsDT0xNjx44VOTHRu0nymIq/vz9CQ0MLLZsyZYryzxEREVAoFJDL5VAoFMrlVlZW2LJlC5YvX46JEydCR0cHNjY28PPzUx5zIZKKH374AfPmzQPwqlwmT56MX3/9FX5+fvD29lau16JFC2zcuBFjx47lXgpJnkyo4Ce9X7p0CR06dBA7Rolo6oG9stCmMb+ezlqyZAlkMhk2bNiAcePGFXr91q1biI2NhaWlJaysrCpMmWjT9/yhNHXMb/vZKck9FSJtJZfLMXHiRKxfvx46OjrYtm0bhg8fXmgdmUyG5s2bo0aNGhr5w4YqNpYKUTnJzc2Fu7s7QkJCYGBggODgYPTv31/sWEQqxVIhKgcZGRlwcXFBeHg4jI2NcfDgQXTr1k3sWEQqx1IhUrOkpCT069cP0dHRMDMzw9GjR9G2bVuxYxGpBUuFSI0eP34MOzs7xMTEoHHjxjh+/DiaNm0qdiwitZHkdSpE2iAmJga2traIiYlBy5Ytce7cORYKaT2WCpEa/Pnnn+jWrRseP36MLl264PTp0zyTiyoElgqRikVGRqJHjx5ISkqCg4MDwsLCin3OD5E2YqkQqdDevXvh4OCAjIwMjBgxAr///juqVq0qdiyicsNSIVKRTZs2YciQIcjLy8OkSZOwbdu2Yh8aR6TNWCpEKvDTTz9h/PjxUCgUWLRoEVatWoVKlfjPiyoenlJMVAaCIGDWrFlYtmwZZDIZVq9ejS+++ELsWESiYakQlVJBQQG8vLywefNm6OrqYuvWrUXu40VU0bBUiEohJycHrq6u+P3332FoaIi9e/eib9++YsciEh1LhaiEXrx4AWdnZ5w6dQo1a9bEoUOHYGNjI3YsIklgqRCVwJMnT+Dg4IDr16+jbt26CAsLQ8uWLcWORSQZPD2F6ANdv34dNjY2uH79OqysrPDHH3+wUIjewFIh+gCnTp1Ct27d8OTJE9ja2uLs2bNo1KiR2LGIJIelQvQewcHBsLOzw4sXLzBw4EAcP34ctWrVEjsWkSSxVIjeYeXKlXB1dVVeJR8cHAxDQ0OxYxFJFkuFqBgKhQIzZ87El19+CUEQ4Ovri1WrVkFHR0fsaESSxrO/iN6Qm5uL0aNHIygoCLq6uti8eTPc3d3FjkWkEVgqRP/y8uVLuLi44MSJE6hWrRr27NmDPn36iB2LSGOwVIj+z9OnT+Ho6Ii///4bderUwZEjR/gseaISYqkQAbh58yb69u2Lx48fw9LSEkePHkXjxo3FjkWkcXigniq8M2fOwNbWVvno33PnzrFQiEqJpUIV2utjJq/v5xUeHo7atWuLHYtIY7FUqMJavXo1hgwZgtzcXEycOBEhISG8BoWojFgqVOEoFArMmjULU6ZMgSAI+OGHH+Dv789rUIhUgAfqqULJzs6Gh4cHQkJCoKuri02bNsHDw0PsWERag6VCFUZiYiI+++wzREVFwdjYGHv27EHv3r3FjkWkVVgqVCHcvHkT/fr1w4MHD9CoUSMcOnQILVq0EDsWkdbhMRXSehEREejSpQsePHiAjh074sKFCywUIjVhqZBW+/XXX9G3b1/l7VdOnjyJOnXqiB2LSGuxVEgrKRQKzJs3D+PGjUNBQQFmzJiBkJAQVKlSRexoRFqNx1RI6+Tk5GD06NHYtWsXdHR04O/vD29vb7FjEVUILBXSKs+fP4ezszPOnz+PatWqITg4GH379hU7FlGFwVIhrREbG4t+/frh3r17aNCgAQ4dOoTWrVuLHYuoQuExFdIKp06dgo2NDe7du4f27dsjKiqKhUIkApYKabxt27ahT58+SE1NxWeffYbTp0+jXr16YsciqpBYKqSxBEGAj48PPDw8kJ+fj6lTp2Lv3r2oWrWq2NGIKiweUyGNlJubi3HjxmHHjh2oVKkSVqxYgcmTJ4sdi6jCY6mQxnn+/DkGDRqEM2fOoGrVqggKCoKTk5PYsYgILBXSMNevX0f//v3x4MED1KtXDwcPHkS7du3EjkVE/0eyx1RiYmLg5eWFDh06oE2bNnB3d0d0dPR7t4uKioKrqyvatGmD9u3bw9vbG/fv3y+HxKRuBw4cgI2NDR48eABra2tER0ezUIgkRpKl8ujRI7i5uSE1NRXLli3DihUrYGRkhLFjx+Lq1atv3S4yMhIeHh7Q19eHn58f/Pz8EB8fj5EjR+L58+flOAJSJUEQsHbtWgwYMAAZGRlwdXXF6dOnUb9+fbGjEdEbJDn9tXbtWsjlcgQEBKBWrVoAAFtbW9jZ2cHPzw9btmwpdrslS5agbt262LhxI/T19QEArVu3Rq9evbBx40bMmTOnvIZAKpKbmwsvLy9s3boVAPDdd99h3rx5kMlkIicjouJIbk9FEASEh4ejS5cuykIBAH19fdjZ2SEqKgppaWlFtktJScHDhw/RpUsXZaEAQM2aNdGjRw+Eh4eXS35SnYSEBPTo0QNbt26FoaEhQkJCMH/+fBYKkYRJrlTi4uKQnp4Oc3PzIq+Zm5tDoVDg9u3bRV7T1X210/XvQnnN1NQUT548QVZWluoDk1pcvXoVnTp1wh9//IGPP/4Y+/btw6BBg8SORUTvIblSSU5OBvBqD+NNr5e9XuffjI2NYWJigsuXLxd57fr16wCA1NRUVUYlNdm3bx9sbW3x6NEjfPLJJ4iOjkbLli3FjkVEH0BypZKXlweg+D0OPT09AK9ubV6cUaNGITY2FkuWLMHz58+RnJyMpUuX4u7duwAAuVyuptSkCoIg4IcffoCLiwsyMzMxcuRInDhxgg/VItIgkjtQX7lyZQBAfn5+kddeF46hoWGx244ZMwYZGRnYtGkTNm/eDB0dHfTr1w8TJkzAjz/++NYHNMXFxakofflIT0/XuMzvk52djZkzZyI0NBQymQxz5szBxIkTkZKSAkA7x/w+HHPFoG1jllypmJiYAIDyh8m/JSUlFVrnTbq6upg2bRo8PT2RkJAAExMTGBsbY8WKFahSpQpq165d7HaadvPBuLg4jcv8LvHx8RgxYgSio6NhZGSEHTt24LPPPiu0jraN+UNwzBWDpo45Pj6+2OWSK5U6deqgZs2aiI2NLfJabGws9PT0YGFh8c73MDIygpGRkfK/L126hFatWvGsIQm6fPkyBgwYgCdPnqBx48bYv38/WrVqJXYsIiolyR1TAQB7e3ucP3++0AWLWVlZCAsLQ/fu3d96F9rvvvsO/fv3L3Ts5NatW7h48SL69++v9txUMiEhIejatSuePHmCrl27Ijo6moVCpOEkWSoTJ06EoaEhvL29cfLkSZw7dw4TJ05EdnY2pk+fDgCIjo5G8+bNsWvXLuV2n3zyCW7fvo0ZM2YgKioKBw4cgLe3N9q2bQsXFxexhkNvUCgU+PbbbzFkyBBkZ2djzJgxCA8Pf+u0JhFpDslNfwGAmZkZAgMDsXTpUkyfPh2CIKBt27bYunUrmjZtCuDVmUJyuRwKhUK5XZ8+feDr64tNmzbB09MTxsbG6Nu3L7788kvldSwkrvT0dHh4eGDfvn2oVKkSli5dimnTpnFqkkhLSPYnbZMmTbBu3bq3vt65c+dij7s4OzvD2dlZndGolO7cuQNnZ2fcvHkTNWrUQGBgIBwcHMSORUQqJMnpL9I+hw8fRseOHXHz5k20aNECFy9eZKEQaSGWCqmVIAj4/vvv4eTkhJcvX2LgwIH4448/lNOYRKRdJDv9RZovIyMDo0ePxp49eyCTybB48WLMnTuXx0+ItBhLhdTi7t27cHZ2xo0bN1C9enXs2LED/fr1EzsWEakZS4VU7ujRoxg+fDhevHgBKysr7Nu3770XrBKRduAxFVIZQRCwZMkSODo64sWLF3B2dsaFCxdYKEQVCEuFVCIjIwPDhg3DnDlzIAgCFi1ahD179sDY2FjsaERUjjj9RWV2//59ODs749q1a6hWrRq2b99e5IaQRFQxsFSoTMLCwuDq6orU1FRYWlpi3759aNasmdixiEgknP6iUhEEAUuXLoWDgwNSU1Ph5OSEqKgoFgpRBcdSoRLLyMjAiBEjMGvWLCgUCixcuBC///47qlevLnY0IhIZp7+oRGJiYjBo0CDcvHkTRkZG2LZtG++1RkRK3FOhDxYSEqK8f5eVlRWio6NZKERUCEuF3is/Px9fffUVhgwZojx1ODo6GlZWVmJHIyKJ4fQXvVN8fDyGDRuGM2fOQFdXFz///DMmT57M+3cRUbFYKvRWp0+fxtChQ5GQkIB69eohODgYtra2YsciIgnj9BcVIQgCli1bhp49eyIhIQGffvopLl++zEIhovdiqVAhaWlpGDJkCGbOnAm5XI7Zs2fj+PHjMDMzEzsaEWkATn+R0o0bNzBw4EDcvn0bxsbG+O2333h2FxGVCPdUCAAQGBiITp064fbt22jZsiX+/PNPFgoRlRhLpYLLy8vD5MmT4ebmhqysLLi7u+PChQswNzcXOxoRaSBOf1VgT548wZAhQ3DhwgXo6elh5cqV8Pb25unCRFRqLJUKKiIiAq6urkhKSsLHH3+MkJAQdOrUSexYRKThOP1VwcjlcixatAh9+vRBUlIS+vTpg8uXL7NQiEgluKdSgTx79gxubm6IjIyETCbDggUL4OPjAx0dHbGjEZGWYKlUEJGRkRgxYgQSEhJgYmKCHTt2oE+fPmLHIiItw+kvLSeXy/HNN9+gd+/eyqvjr1y5wkIhIrXgnooWe/bsGUaMGIETJ05AJpNh4cKFWLhwIae7iEhtWCpaKiIiAm5ubkhISICpqSl27NiB3r17ix2LiLQcp7+0jFwuh4+PD/r06VNououFQkTlgXsqWiQ+Ph5ubm7K6S4fHx8sWLCA011EVG5YKloiPDwcbm5uSExMhKmpKQIDA9GrVy+xYxFRBcPpLw33errLzs4OiYmJ6NGjB65cucJCISJRcE9Fg8XHx2PEiBE4efIkp7uISBJYKhrq+PHjcHd3R2JiIszMzLBjxw7unRCR6Dj9pWHy8/Pxww8/wN7eHomJiejZsyenu4hIMlgqGuTevXuwtbXFmjVrIJPJ8M033yAsLAx16tQROxoREQBOf2mM7du3Y+LEiUhPT0f9+vURFBSErl27ih2LiKgQ7qlIXHp6Ojw8PDBy5Eikp6dj8ODBCAsLY6EQkSSxVCTs4sWLaNeuHbZt2wZDQ0Ns2LABwcHBqFGjhtjRiIiKxVKRIIVCgZ9++gldunTBvXv30KZNG1y6dAnjx4/no36JSNJ4TEVi4uPj4eHhgfDwcADAlClT4OvrCwMDA5GTERG9H0tFQg4dOoTRo0cjKSkJH330EbZs2YJ+/fqJHYuI6INx+ksCcnJyMHXqVDg5OSEpKQm9e/fG33//zUIhIo0j2VKJiYmBl5cXOnTogDZt2sDd3R3R0dHv3EYQBISEhGDw4MGwtrZGmzZtMHDgQOzevbucUpfcrVu38Mknn2DVqlXQ1dWFr68vjh07hrp164odjYioxCRZKo8ePYKbmxtSU1OxbNkyrFixAkZGRhg7diyuXr361u3WrFmDefPmoXXr1li9ejXWrFkDc3NzzJ8/Hxs2bCjHEbyfIAjYuHEjOnTogKtXr6JJkyY4f/48Zs2ahUqVJPm1EBG9lySPqaxduxZyuRwBAQGoVasWAMDW1hZ2dnbw8/PDli1bit0uKCgI7dq1w8KFC5XLbG1tcenSJRw8eBCenp7lEf+9UlJSMGHCBISEhAAARo4ciTVr1qBatWoiJyMiKhvJ/UosCALCw8PRpUsXZaEAgL6+Puzs7BAVFYW0tLRit1UoFKhSpUqhZTKZDFWrVlVr5pI4fvw4WrVqhZCQEFSrVg3bt2/H1q1bWShEpBUkVypxcXFIT0+Hubl5kdfMzc2hUChw+/btYrcdM2YM/vjjD+zevRsvX75EVlYWdu7cidjYWIwaNUrd0d8pOzsbX375Jezs7BAXF4cuXbrgypUrcHNzEzUXEZEqSW76Kzk5GQBQs2bNIq+9XvZ6nTd5enqiSpUqWLRoEebPnw8AMDQ0hK+vLwYMGKCmxO/3ujxu3rwJXV1dfPPNN5g9ezZ0dSX3109EVCaS+6mWl5cH4NV015v09PQAvDoFtzinTp2Cr68vHBwc8Nlnn6GgoAD79u3DwoULUbNmTXTv3l19wYshl8vx888/Y/78+cjPz4elpSW2b98Oa2vrcs1BRFReJFcqlStXBvDquSFvel04hoaGxb42b948tGvXDkuXLlUu79GjBwYNGoRvv/1WeZX6m+Li4lQRvZAnT55g6tSpuHDhAgBg1KhRWLBgAQwNDcv8eenp6WrJLGUcc8XAMWs+yZWKiYkJgFdnSL0pKSmp0Dr/9uDBAzx//hyjR48u8lqnTp3w66+/Ijk5GbVr1y7yer169cqY+v8TBAHbt2/HpEmTkJaWBjMzM/z6669wdHRU2WfExcWpNLMm4JgrBo5Zc8THxxe7XHKlUqdOHdSsWROxsbFFXouNjYWenh4sLCyKvCYIAgCgoKCgyGuv93rUff1HSkoKPv/8cwQHBwMAnJ2dsX79+mJLkIhIG0nu7C8AsLe3x/nz5/H8+XPlsqysLISFhaF79+7FniL8n//8BwYGBjh37lyR16Kjo2FqalrswX9VCQ8PR+vWrREcHAwjIyNs2rQJe/fuZaEQUYUiyVKZOHEiDA0N4e3tjZMnT+LcuXOYOHEisrOzMX36dACviqJ58+bYtWsXgFcH9j09PREdHY3Zs2fj9OnTOHXqFKZOnYrY2Fh4e3urJWtOTg6mTZuGPn364OnTp7CxscGVK1cwduxY3qaeiCocyU1/AYCZmRkCAwOxdOlSTJ8+HYIgoG3btti6dSuaNm0K4NV0l1wuh0KhUG43adIkNGnSBAEBAThy5AhkMhnMzc3h5+en0mMar129ehVubm64ceMGdHV14ePjg6+//pqnChNRhSXZn35NmjTBunXr3vp6586diz3u4uDgAAcHB3VGA/DqPmPTpk1Dfn4+LCwssH37dnTs2FHtn0tEJGWSnP6SuoKCAkyfPh35+fn4/PPPcfnyZRYKEREkvKciZbq6uti/fz+qVq2Krl27ih2HiEgyWCqlZG9vL3YEIiLJ4fQXERGpDEuFiIhUhqVCREQqw1IhIiKVYakQEZHKsFSIiEhlWCpERKQyLBUiIlIZlgoREakMS4WIiFSGpUJERCrDUiEiIpWRCa8f7l5BXbp0SewIREQaqUOHDkWWVfhSISIi1eH0FxERqQxLhYiIVIalImFyuRy//PIL7Ozs0LJlS/zvf/+Dr68vcnJySvxeo0aNgqWlJaKiotSQVHXKMub09HT4+vqib9++aNOmDezs7LB48WKkpqaWQ/IPFxMTAy8vL3To0AFt2rSBu7s7oqOj1badFJQ2+/nz5zFy5Eh07twZnTt3hpubG06dOlUOictOFd/XxYsX0axZM4wcOVJNKVWPpSJhixcvhr+/P1xcXLBlyxaMGzcOu3btwrRp00r0Pnv37sWFCxfUlFK1Sjvm/Px8jB8/Hrt378aIESOwceNGDB8+HHv27MHnn38OhUJRTiN4t0ePHsHNzQ2pqalYtmwZVqxYASMjI4wdOxZXr15V+XZSUNrsERERGDNmDAwMDLB06VL89NNP0NfXh5eXF06fPl2OIyg5VXxfeXl5WLBgATTusLdAkvTo0SPB0tJS8PX1LbR8x44dgoWFhXDx4sUPep/k5GShU6dOgpeXl2BhYSFcuHBBHXFVoixjPnr0qGBhYSHs3bu30PKAgADBwsJCOH/+vFoyl9Ts2bOFNm3aCMnJycplubm5wv/+9z9h1KhRKt9OCkqb3cnJSbCzsxPy8vKUy7KysoROnToJw4cPV2fkMlPF9+Xn5yfY2NgIAwcOFNzd3dWUVPW4pyJREREREAQB/fv3L7S8X79+kMlkCA8P/6D3+f7771GjRg24u7urI6ZKlWXMTZs2xffff48+ffoUWt6sWTMAQHx8vOoDl5AgCAgPD0eXLl1Qq1Yt5XJ9fX3Y2dkhKioKaWlpKttOCsoy5s8//xyLFi2Cnp6ecrmhoSEaNWqEZ8+elUv+0lDF93Xnzh1s3LgRX331FapUqaLuyCrFUpGo2NhYyGQyNGnSpNDy6tWrw9TUFLdu3Xrve5w+fRoHDx7EokWLoK+vr66oKlOWMTdp0gSDBw+GkZFRoeX3798HADRs2FD1gUsoLi4O6enpMDc3L/Kaubk5FAoFbt++rbLtpKC02WUyGRwdHfHJJ58UWp6fn4+HDx9K4vt8m7J+XwqFAgsWLEC7du0waNAgdUZVC5aKRCUnJ8PIyKjYMqhZsyZSUlLeuX1WVha++eYbuLi4FPmHKVVlHfObEhISsG7dOlhaWhZ7kVZ5S05OBvBqLG96vez1OqrYTgpUnX316tV48eIFRowYoZqAalDWMe/cuRM3btzAokWL1BNQzXTFDlDR5Ofn49GjR+9cp1q1asjNzX3r3oWenh5evnz5zvdYuXIlMjMzMWvWrFJnVZXyGvO/JScnw9vbG3l5eVi2bBlkMlmJMqtDXl4eABQ7xtdTPMWd5Vba7aRAldmDgoKwfv16DBw4EHZ2dqoLqWJlGfOzZ8/w888/w8vLC//973/VF1KNWCrlLCEhAY6Oju9cx8XFBQYGBsjPzy/29by8PBgYGLx1+2vXrmHbtm1YvHhxoTldsZTHmP/t0aNHGD9+PFJTU7Fx40ZYWFiUOLM6VK5cGQCKHePrH0SGhoYq204KVJXd398fq1evRv/+/fHdd9+pNqSKlWXM3377LUxMTDBhwgT1BVQzlko5a9CgAWJjY9+73rx585Ceno68vLwiv/EkJycrD0C/qaCgQDkfa29vj8zMTAD//zejnJwcZGZmomrVqmUcyYdT95j/LSYmBmPHjoWBgQF27tyJpk2bljq3qpmYmABAsdN4SUlJhdZRxXZSoIrsPj4+CAoKwvjx4zFjxgxJ7HW+S2nHfOzYMURGRiIgIAD5+fnKUpLL5QCAzMxM6OnpSf74KEtFoqysrCAIAu7cuYMWLVoolyclJSEpKanQsn979uyZ8oB2+/bti7zu5eUFAB/0Q768lXbMr8XFxWH8+PGoXbs2Nm7cCDMzM3VHLpE6deqgZs2axf7dx8bGQk9Pr9i9qtJuJwVlze7n54ddu3Zh3rx58PDwUGdUlSntmE+cOAFBEJT/Rt/Uvn17TJo0CZMnT1Z5ZlViqUhUnz598OOPP+L3338v9MP0999/BwA4ODgUu52pqSl27NhRZPmtW7ewePFizJ8/H1ZWVuoJXUalHTPw6re5yZMnQ19fH5s3b8ZHH32k9rylYW9vj9DQUDx//lz522pWVhbCwsLQvXv3t+5BlnY7KSht9vDwcKxbtw4zZszQmEJ5rTRj9vb2xuDBg4ssX7x4MQBg/vz5qFevnnqDq4JYF8jQ+y1btkywsrIS1q5dK/z555/Cb7/9JrRu3Vr4+uuvC63n4eEhdOvW7Z3vdeHCBclf/CgIpR/z7t27BQsLC2HdunXC33//XeR/jx8/Lu+hFOvZs2dCp06dhIEDBwonTpwQzp49K4waNUpo27atcOfOHUEQBCEqKkqwsrISgoKCSrSdVJVmzPn5+UKvXr2Enj17ClevXi32O83NzRVzWO9U2u+5OO7u7hp18SP3VCRs+vTpqF69OoKDg7FmzRqYmppi3Lhx+Pzzzwutp1AolPOumq60Y/7zzz8BAMuXL8fy5cuLvK+LiwuWLFmi3vAfwMzMDIGBgVi6dCmmT58ORmAHUAAAArVJREFUQRDQtm1bbN26VXn8RxAEyOXyQreW+ZDtpKo0Y3727BkeP34MABgyZEix7xsREYEGDRqUzyBKqLTfszbg81SIiEhlePEjERGpDEuFiIhUhqVCREQqw1IhIiKVYakQEZHKsFSIiEhlWCpERKQyLBUiIlIZlgoREakMS4WIiFSGpUJERCrDUiEiIpVhqRARkcqwVIiISGVYKkREpDJ8SBeRhJw8eRK//fYb8vLyIJfLMWfOHLRp0wZLly7FlStXALx64Fhxj50lkgI+pItIIkJDQ3HmzBl8//33MDQ0xJYtW7BmzRoMGjQIrVu3Ru/evfHFF1/gzJkzuHDhAmrUqCF2ZKIiOP1FJAEJCQk4evQofvrpJxgaGgIAmjZtirS0NDx69AiOjo64f/8+Tp8+jYYNG8LIyEjkxETF4/QXkQTs378fnp6e0NX9//8k7969CwDo168fAKBZs2Y4cOAA6tatW2g9Iinh9BeRBAiCAJlMVmiZt7c3Tp48iXPnzqF27doiJSMqGU5/EUnAm4WSn5+P6OhomJubs1BIo7BUiCToypUryMzMhI2NTZHXOLlAUsZSIZKA5ORkPHz4UPnf586dAwB06tSp0HrBwcEICwsr12xEJcFSIRLZy5cv4eTkhL59++Lly5cQBAFHjx4FAJibmyvXy8jIwIEDB9CjRw+xohK9F0uFSGRPnz5FSkoKPvvsMxgbG2Pt2rXo2rUrdHV1cfv2bQBASkoKpk2bhqlTp0JfX1/kxERvx7O/iCTA398fUVFRKCgowKeffooJEyYgMjIS/v7+0NfXh76+PqZMmQJra2uxoxK9E0uFiIhUhtNfRESkMiwVIiJSGZYKERGpDEuFiIhUhqVCREQqw1IhIiKVYakQEZHKsFSIiEhlWCpERKQyLBUiIlIZlgoREanM/wNYaxcTBj9+QQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = lambda x : (np.exp(x) - 1) / x\n",
    "x = np.linspace(-0.5, 0.5, 20)\n",
    "\n",
    "plt.figure(figsize=(6,6))\n",
    "plt.plot(x, f(x), 'k')\n",
    "plt.plot(0, 1, 'o',   markersize=10, markerfacecolor='w',\n",
    "         markeredgewidth=1.5, markeredgecolor='k')\n",
    "plt.text(-0.4, 1.1, r\"$y=\\frac{e^x - 1}{x}$\", fontsize=30)\n",
    "plt.xlabel(r'$x$', fontsize=25)\n",
    "plt.ylabel(r'$y$', fontsize=25)\n",
    "plt.xticks(fontsize=18)\n",
    "plt.yticks(fontsize=18)a\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "주어진 함수 $f(x)$는 $x=0$에서 정의되지 않지만 예상처럼 그래프를 통해 $x \\to 0$일 때 $f(x) \\to 1$인 것을 알 수 있습니다. 이 극한값을 구하기 위해서는 약간의 대수적 조작이 필요합니다. \n",
    "\n",
    "$$\\lim_{x \\to 0} \\frac{e^x - 1}{x} $$\n",
    "\n",
    "위 식에서 분자 $e^x - 1$를 $t$로 치환합니다. $t=e^x -1$으로 두면 $e^x = 1+t$입니다. 여기서 $x$를 로그로 표현하면 $x = \\ln (1+t)$입니다. 한편 $x \\to 0$일 때 $t$ 역시 0으로 다가가므로 주어진 함수를 다음처럼 바꿔 쓸 수 있습니다.\n",
    "\n",
    "$$\n",
    "\\lim_{t \\to 0} \\frac{t}{\\ln(1+t)}\n",
    "$$\n",
    "\n",
    "이제 다음과 같은 과정으로 극한을 구할 수 있습니다.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\lim_{t \\to 0} \\frac{t}{\\ln(1+t)} \n",
    "&= \\lim_{t \\to 0} \\frac{1}{ \\dfrac{\\ln(1+t)}{t} } \\\\[5pt]\n",
    "&= \\lim_{t \\to 0} \\frac{1}{ \\dfrac{1}{t}\\ln(1+t)  } \\\\[5pt]\n",
    "&= \\lim_{t \\to 0} \\frac{1}{\\ln(1+t)^{\\frac{1}{t}} } \n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "마지막 등호는 로그의 성질 $\\log_{a} M^k = k \\log_{a} M$를 이용하였습니다. 분모에서 $(1+t)^{\\frac{1}{t}} $ 부분은 ${t \\to 0}$일 때 $e$입니다.($e$의 정의) 따라서\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\lim_{t \\to 0} \\frac{t}{\\ln(1+t)} \n",
    "&= \\lim_{t \\to 0} \\frac{1}{ \\dfrac{\\ln(1+t)}{t} } \\\\[5pt]\n",
    "&= \\lim_{t \\to 0} \\frac{1}{\\ln(1+t)^{\\frac{1}{t}} } \\\\[5pt]\n",
    "&=  \\frac{1} {\\ln e } = 1\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "<hr/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. $\\lim_{x \\to 0} \\dfrac{a^x - 1}{x} = \\ln a$\n",
    "\n",
    "2번도 비슷하게 1번 과정과 비슷하게 극한값을 구할 수 있습니다. 극한값을 그래프로 확인해보기 위해 먼저 그래프를 그려봅시다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = np.e**2\n",
    "f = lambda x : (a**x - 1) / x\n",
    "x = np.linspace(-0.5, 0.5, 20)\n",
    "\n",
    "plt.figure(figsize=(6,6))\n",
    "plt.plot(x, f(x), 'k')\n",
    "plt.plot(0, np.log(a), 'o',    markersize=10, markerfacecolor='w',\n",
    "         markeredgewidth=1.5, markeredgecolor='k')\n",
    "plt.text(-0.4, 3, r\"$y=\\frac{a^x - 1}{x}, \\quad a=e^2$\", fontsize=30)\n",
    "plt.xlabel(r'$x$', fontsize=25)\n",
    "plt.ylabel(r'$y$', fontsize=25)\n",
    "plt.xticks(fontsize=18)\n",
    "plt.yticks(fontsize=18)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "그래프를 그리기 위해 $a=e^2$로 설정했습니다. $x \\to 0$일 때 함숫값이 2로 수렴합니다. $\\ln e^2 = 2$이므로 주어진 함수의 극한값은 $\\ln a$가 맞는 듯 보입니다. 구체적인 계산은 1번 과정과 완전히 동일합니다.\n",
    "\n",
    "\n",
    "$$\\lim_{x \\to 0} \\frac{a^x - 1}{x} $$\n",
    "\n",
    "위 식에서 분자 $a^x - 1$를 $t$로 치환합니다. $t=a^x -1$으로 두면 $a^x = 1+t$입니다. 여기서 $x$를 로그로 표현하면 $x = \\log_a (1+t)$입니다. 한편 $x \\to 0$일 때 $t$ 역시 0으로 다가가므로 주어진 함수를 다음처럼 바꿔 쓸 수 있습니다.\n",
    "\n",
    "$$\n",
    "\\lim_{t \\to 0} \\frac{t}{\\log_a(1+t)}\n",
    "$$\n",
    "\n",
    "이제 다음과 같은 과정으로 극한을 구할 수 있습니다.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\lim_{t \\to 0} \\frac{t}{\\log_a(1+t)} \n",
    "&= \\lim_{t \\to 0} \\frac{1}{ \\dfrac{\\log_a(1+t)}{t} } \\\\[5pt]\n",
    "&= \\lim_{t \\to 0} \\frac{1}{\\log_a(1+t)^{\\frac{1}{t}} } \n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "분모에서 $(1+t)^{\\frac{1}{t}} $ 부분은 ${t \\to 0}$일 때 $e$입니다.($e$의 정의) 따라서\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\lim_{t \\to 0} \\frac{t}{\\log_a(1+t)} \n",
    "&= \\lim_{t \\to 0} \\frac{1}{ \\dfrac{\\log_a(1+t)}{t} } \\\\[5pt]\n",
    "&= \\lim_{t \\to 0} \\frac{1}{\\log_a(1+t)^{\\frac{1}{t}} } \\\\[5pt]\n",
    "&=  \\frac{1} {\\log_a e } = \\ln a\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "마지막 등호에서 로그의 밑변환 성질(3장 로그의 성질)을 사용했습니다.\n",
    "\n",
    "$$\n",
    "\\frac{1} {\\log_a e } = \\frac{1}{\\dfrac{\\log_e e}{\\log_e a}} = \\log_e a = \\ln a\n",
    "$$\n",
    "<hr/>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 로그함수의 극한"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. $\\lim_{x \\to 0} \\dfrac{\\ln(1+x)}{x} = 1$\n",
    "\n",
    "이 경우는 위에서 알아본 지수함수의 극한을 계산하는 과정을 이용하면 간단하게 구할 수 있습니다. 계산을 하기 앞서 이번에도 그림으로 먼저 확인해보겠습니다\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = lambda x : np.log(1+x) / x\n",
    "x = np.linspace(-0.5, 0.5, 20)\n",
    "\n",
    "plt.figure(figsize=(6,6))\n",
    "plt.plot(x, f(x), 'k')\n",
    "plt.plot(0, 1, 'o',    markersize=10, markerfacecolor='w',\n",
    "         markeredgewidth=1.5, markeredgecolor='k')\n",
    "plt.text(0., 1.1, r\"$y=\\frac{\\ln (1+x)}{x}$\", fontsize=30)\n",
    "plt.xlabel(r'$x$', fontsize=25)\n",
    "plt.ylabel(r'$y$', fontsize=25)\n",
    "plt.xticks(fontsize=18)\n",
    "plt.yticks(fontsize=18)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$x \\to 0$일 때 극한값 1을 가짐을 확인할 수 있습니다. 구체적인 계산 과정은 아래와 같습니다.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\lim_{x \\to 0} \\frac{\\ln(1+x)}{x} \n",
    "&= \\lim_{x \\to 0} \\frac{1}{x} \\ln(1+x) \\\\[5pt]\n",
    "&=  \\lim_{x \\to 0} \\ln(1+x)^{\\frac{1}{x} } \\\\[5pt]\n",
    "&= \\ln e = 1\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4. $\\lim_{x \\to 0} \\dfrac{\\log_a(1+x)}{x} = \\dfrac{1}{\\ln a}$\n",
    "\n",
    "이 경우도 동일하게 보일 수 있습니다. 우선 그림을 그려보겠습니다.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = np.e**2\n",
    "f = lambda x : (np.log(1+x)/np.log(a)) / x\n",
    "x = np.linspace(-0.5, 0.5, 20)\n",
    "\n",
    "plt.figure(figsize=(6,6))\n",
    "plt.plot(x, f(x), 'k')\n",
    "plt.plot(0, 1/np.log(a), 'o',   markersize=10, markerfacecolor='w',\n",
    "         markeredgewidth=1.5, markeredgecolor='k')\n",
    "plt.text(-0.4, 0.65, r\"$y=\\frac{\\log_a (1+x)}{x},\\quad a=e^2$\", fontsize=30)\n",
    "plt.xlabel(r'$x$', fontsize=25)\n",
    "plt.ylabel(r'$y$', fontsize=25)\n",
    "plt.xticks(fontsize=18)\n",
    "plt.yticks(fontsize=18)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "이번에도 $a=e^2$로 두고 그림을 그려보면 극한값이 $\\dfrac{1}{2}$이라는 것을 알 수 있습니다. 주어진 함수의 극한값이 $\\dfrac{1}{\\ln a}$라면 이 경우 $\\dfrac{1}{\\ln e^2}= \\dfrac{1}{2}$가 되어 그림의 결과와 일치하는 것을 알 수 있습니다. 구체적으로 계산해보면 다음과 같습니다.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\lim_{x \\to 0} \\frac{\\log_a(1+x)}{x} \n",
    "&= \\lim_{x \\to 0} \\frac{1}{x} \\log_a(1+x) \\\\[5pt]\n",
    "&= \\lim_{x \\to 0}\\log_a(1+x) ^ {\\frac{1}{x}} \\\\[5pt]\n",
    "&= \\log_a e = \\frac{1}{\\ln a}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "이번에도 마지막 등호에서 로그의 밑변환 성질을 이용했습니다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "위 4가지 극한은 지수, 로그 함수의 도함수 공식을 유도할 때 유용하게 사용되므로 잘 알아두면 나중에 도함수의 공식이 기억이 나지 않을 때공식을 직접 유도할 수 있기 때문에 많은 도움이 됩니다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<link href='https://fonts.googleapis.com/earlyaccess/notosanskr.css' rel='stylesheet' type='text/css'>\n",
       "<!--https://github.com/kattergil/NotoSerifKR-Web/stargazers-->\n",
       "<link href='https://cdn.rawgit.com/kattergil/NotoSerifKR-Web/5e08423b/stylesheet/NotoSerif-Web.css' rel='stylesheet' type='text/css'>\n",
       "<!--https://github.com/Joungkyun/font-d2coding-->\n",
       "<link href=\"http://cdn.jsdelivr.net/gh/joungkyun/font-d2coding/d2coding.css\" rel=\"stylesheet\" type=\"text/css\">\n",
       "<style>\n",
       "    h1 { font-family: 'Noto Sans KR' !important; color:#348ABD !important;   }\n",
       "    h2 { font-family: 'Noto Sans KR' !important; color:#467821 !important;   }\n",
       "    h3 { font-family: 'Noto Sans KR' !important; color:#A60628 !important;   }\n",
       "    h4 { font-family: 'Noto Sans KR' !important; color:#7A68A6 !important;   }        \n",
       "            \n",
       "    p:not(.navbar-text) { font-family: 'Noto Serif KR', 'Nanum Myeongjo'; font-size: 11pt; line-height: 200%;  text-indent: 10px; }\n",
       "    li:not(.dropdown):not(.p-TabBar-tab):not(.p-MenuBar-item):not(.jp-DirListing-item):not(.p-CommandPalette-header):not(.p-CommandPalette-item):not(.jp-RunningSessions-item):not(.p-Menu-item)   \n",
       "            { font-family: 'Noto Serif KR', 'Nanum Myeongjo'; font-size: 12pt; line-height: 200%; }\n",
       "    table  { font-family: 'Noto Sans KR' !important;  font-size: 11pt !important; }        \n",
       "    li > p  { text-indent: 0px; }\n",
       "    li > ul { margin-top: 0px !important; }       \n",
       "    sup { font-family: 'Noto Sans KR'; font-size: 9pt; } \n",
       "    code, pre  { font-family: D2Coding, 'D2 coding' !important; font-size: 11pt !important; line-height: 130% !important;}\n",
       "    .code-body { font-family: D2Coding, 'D2 coding' !important; font-size: 11pt !important;}\n",
       "    .ns        { font-family: 'Noto Sans KR'; font-size: 15pt;}\n",
       "    .summary   {\n",
       "                   font-family: 'Georgia'; font-size: 12pt; line-height: 200%; \n",
       "                   border-left:3px solid #D55E00; \n",
       "                   padding-left:20px; \n",
       "                   margin-top:10px;\n",
       "                   margin-left:15px;\n",
       "               }\n",
       "    .green { color:#467821 !important; }\n",
       "    .comment { font-family: 'Noto Sans KR'; font-size: 10pt; }\n",
       "</style>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
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     "output_type": "display_data"
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   ],
   "source": [
    "%%html\n",
    "<link href='https://fonts.googleapis.com/earlyaccess/notosanskr.css' rel='stylesheet' type='text/css'>\n",
    "<!--https://github.com/kattergil/NotoSerifKR-Web/stargazers-->\n",
    "<link href='https://cdn.rawgit.com/kattergil/NotoSerifKR-Web/5e08423b/stylesheet/NotoSerif-Web.css' rel='stylesheet' type='text/css'>\n",
    "<!--https://github.com/Joungkyun/font-d2coding-->\n",
    "<link href=\"http://cdn.jsdelivr.net/gh/joungkyun/font-d2coding/d2coding.css\" rel=\"stylesheet\" type=\"text/css\">\n",
    "<style>\n",
    "    h1 { font-family: 'Noto Sans KR' !important; color:#348ABD !important;   }\n",
    "    h2 { font-family: 'Noto Sans KR' !important; color:#467821 !important;   }\n",
    "    h3 { font-family: 'Noto Sans KR' !important; color:#A60628 !important;   }\n",
    "    h4 { font-family: 'Noto Sans KR' !important; color:#7A68A6 !important;   }        \n",
    "            \n",
    "    p:not(.navbar-text) { font-family: 'Noto Serif KR', 'Nanum Myeongjo'; font-size: 11pt; line-height: 200%;  text-indent: 10px; }\n",
    "    li:not(.dropdown):not(.p-TabBar-tab):not(.p-MenuBar-item):not(.jp-DirListing-item):not(.p-CommandPalette-header):not(.p-CommandPalette-item):not(.jp-RunningSessions-item):not(.p-Menu-item)   \n",
    "            { font-family: 'Noto Serif KR', 'Nanum Myeongjo'; font-size: 12pt; line-height: 200%; }\n",
    "    table  { font-family: 'Noto Sans KR' !important;  font-size: 11pt !important; }        \n",
    "    li > p  { text-indent: 0px; }\n",
    "    li > ul { margin-top: 0px !important; }       \n",
    "    sup { font-family: 'Noto Sans KR'; font-size: 9pt; } \n",
    "    code, pre  { font-family: D2Coding, 'D2 coding' !important; font-size: 11pt !important; line-height: 130% !important;}\n",
    "    .code-body { font-family: D2Coding, 'D2 coding' !important; font-size: 11pt !important;}\n",
    "    .ns        { font-family: 'Noto Sans KR'; font-size: 15pt;}\n",
    "    .summary   {\n",
    "                   font-family: 'Georgia'; font-size: 12pt; line-height: 200%; \n",
    "                   border-left:3px solid #D55E00; \n",
    "                   padding-left:20px; \n",
    "                   margin-top:10px;\n",
    "                   margin-left:15px;\n",
    "               }\n",
    "    .green { color:#467821 !important; }\n",
    "    .comment { font-family: 'Noto Sans KR'; font-size: 10pt; }\n",
    "</style>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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