mathanalysis_practice106.ipynb

{
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   "source": [
    "%%html\n",
    "<style>\n",
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ссылки / Links\n",
    "[Математический анализ. Первый семестр.](http://lectoriy.mipt.ru/course/Maths-Basic_Analysis/lectures)\n",
    "\n",
    "[Непрерывность](https://ru.coursera.org/lecture/matematika-dlya-vseh/opriedielieniie-priediela-1-xzyQ8)\n",
    "\n",
    "[Непрерывность функции. Основные элементарные функции. Замечательные пределы.](https://www.youtube.com/watch?v=0J4Wm_4YKdw&t=57s)\n",
    "\n",
    "[Пределы (презентация)](https://en.ppt-online.org/101123)\n",
    "\n",
    "[Примеры решения задач](http://ru.solverbook.com/primery-reshenij/)\n",
    "\n",
    "[Примеры решения с помощью замечательных пределов](http://ru.solverbook.com/primery-reshenij/primery-resheniya-s-pomoshhyu-zamechatelnyx-predelov/)\n",
    "\n",
    "[Примеры решения: второй замечательный предел](http://ru.solverbook.com/primery-reshenij/primery-resheniya-vtoroj-zamechatelnyj-predel/)\n",
    "\n",
    "---\n",
    "\n",
    "[Limits and continuity](https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new)\n",
    "\n",
    "[SymPy Calculus](https://docs.sympy.org/latest/tutorial/calculus.html)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Задачи / Problems\n",
    "[Сборник задач по математике для втузов. В 4-х частях. Под ред. Ефимова А.В., Поспелова А.С. (Часть 2)](https://drive.google.com/file/d/1qUN5t1IQdWvlRf0gdVYZ87MznWUaHq2x/view)\n",
    "\n",
    "![](math_images/ma106_01.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math, sympy\n",
    "from sympy import *\n",
    "sympy.init_printing(use_unicode=True)\n",
    "import pylab as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.305\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} x \\ ctg \\ \\pi x}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAAqBAMAAACabMzBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMt0Qq5nNdrvviURUZiKjDJpDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAYElEQVQYGWNgYGAQMgASDMyu+WCagaGeNrTHfJUCkEWUgf9g8IEyQ7DprgQa/JWBgWv1KdlCAQYGGYY6diAFBA0sBiCK0YEJRDGwHuAH0/YG/BtAjDIGzgUguoeBBUgDAFkwHZVFxTXoAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{\\pi}$$"
      ],
      "text/plain": [
       "1\n",
       "─\n",
       "π"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit( x * cot(pi * x), x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.308\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\frac {\\cos \\alpha x - \\cos \\beta x}{x^2}, \\alpha \\neq \\beta}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{a^{2}}{2} + \\frac{b^{2}}{2}$$"
      ],
      "text/plain": [
       "   2    2\n",
       "  a    b \n",
       "- ── + ──\n",
       "  2    2 "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, a, b = symbols('x a b')\n",
    "limit( (cos(a * x) - cos(b * x)) / (x ** 2), x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.309\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} (\\frac {1}{\\sin x} -  ctg \\ x)}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXaZIs1UEN1mu6tEMu+iw/3TAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAV0lEQVQIHWNgYBQyYWBgCGOocGDgbGBgXMDAtYGB7TsDvwEDzzeGeiD5k8FfgIH3E5j8yFAvwMDzCSTL9o2BawED53cGzgcMzAEMDG0MdRcYGAqV9jIAAMSUErnbiJtLAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit( 1 / sin(x) - cot(x), x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.321\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} \\bigg( \\frac {x^2 + 5}{x^2 - 5} \\bigg) ^ {x^2}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAATBAMAAAB8awA1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mUTdMiJmu6tUze/kkN0jAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAgklEQVQYGWNgAAPJCwwMSkYQNgO7cQIDkwHrAiiXJ4GBr4DhB4LX38DwBsGrF2DIROE9RvCAKv8geKimAG0AWgoCHLpfDBiUTAQYGBiVQSQMhBWwK8DYDBw/GZgD4Dyuz0aqcA4D/wYEm4GBXwGZxwbkHYALcF5gEGmA8xiMjA3gHABAjxmYNDN2xwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$e^{10}$$"
      ],
      "text/plain": [
       " 10\n",
       "ℯ  "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit( ((x ** 2 + 5) / (x ** 2 - 5)) ** (x **2), x, oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.325\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\frac {1}{x} \\ln{\\sqrt{\\frac{1 + x}{1 - x}}}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOBAMAAAAoFKpzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMt0Qq5nNdrvviSJUZkQZkPbxAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdklEQVQIHWNgYFRmAINVpRsYGEzCPoM5bAeYtYEMdgjvEgPDCgSvhYHhvAFc7isDw/sDMB4ziHcBxuP+xcAgX4DCWwDjMQPl3sPlGID6zsNNYehgYLiPsOEmA0MY1Hb7BAamA8wTGRhYOj+5MzAFMDAvLxFgAACKdSe0qRxCpwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.0$$"
      ],
      "text/plain": [
       "1.00000000000000"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit( ln(((1 + x) / (1 - x)) ** 0.5) / x, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.326\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} x (a ^ {\\frac{1}{x}} - 1)}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAVBAMAAAAQkWtIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMs3vq927IhCZdkSJVGavtXQaAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABWElEQVQoFWWPPUjDUBSFTxLzZ5ogDjpaNxf/wE0oLi66dBBEQQlSKAhCBh0chCwKTu3gZpF0EjqJP4tThCxChyw6CWZSdNEiSgRB70vyKtEHl3vO+e59eQH+HSdNspbHvWHqL6n1T+UZ7jKfDC3koVbkfoREhZu06xH3TyT+QIkzSG4OsgeedqEYJXC/dQicB883BBpUauPYAwozDAqLUDwhRskgMEZ15igxYBUZlH0InZ4yZinHCqCVoJcBk6qCIfrtLw6nAT1m8xBeGbw/AT6MGKtsk6Bko0aRMcE3P7WDICKGUaDm44VUeq1Uh9ExaZaddYIh1kwHls02hUmIvjE86DIYAKJnLRXo0x4G3qrYbs8D49/vHsELeu3c1WYTkB02nBwl1HaWScl9WYAtLoBbkg9UVp1nR1wAu7SxwWw1y1Q7E9S0dus6uXEvyxT3F3aVGqXyEfgBBupPRvlHYg8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\log{\\left (a \\right )}$$"
      ],
      "text/plain": [
       "log(a)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, a = symbols('x a')\n",
    "limit(x * (a ** (1 / x) - 1), x, oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.327\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 1} \\frac{a ^ x - a}{x - 1}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$a \\log{\\left (a \\right )}$$"
      ],
      "text/plain": [
       "a⋅log(a)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, a = symbols('x a')\n",
    "limit((a ** x - a) / (x - 1), x, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.328\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to a} \\frac{\\log_a{x} - 1}{x - a}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{a \\log{\\left (a \\right )}}$$"
      ],
      "text/plain": [
       "   1    \n",
       "────────\n",
       "a⋅log(a)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, a = symbols('x a')\n",
    "limit((log(x, a) - 1) / (x - a), x, a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.333\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\bigg( \\frac{\\sin x}{x} \\bigg) ^ {\\frac{\\sin x}{x - \\sin x}}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAATBAMAAAB1gKBPAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEIl2mUTdMiJmu6tUzZF4WEUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABdSURBVBgZY2BAAoxzkThAZrEZKp/hGJl89lVAsIKBVP1N95wZGBiVTQSQrQ0rYFdA4nO8ZGAOQOJzPTZSReIy8G1A5jEw8Cmg8tmA/ANIQpwXGEQakPgMRsYGSFwArY8TXU6Jz1oAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$e^{-1}$$"
      ],
      "text/plain": [
       " -1\n",
       "ℯ  "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit( (sin(x) / x) ** (sin(x) / (x - sin(x))), x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.339\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 2\\pm 0} \\frac { 2 + x }{4 - x ^ 2} }}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAAKBAMAAADMVPqUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnaZZruJ70RUIqsKEj5BAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAgElEQVQIHWNggAHXdCBLPF0AxmdgYG+dtZqBR1tkCkLoGgOj/oZzDAycCKEJDAxSfxQYGFg2MAgZg4AJQwADA/f/DUAlBXBlQCGG+Q+ABIgBASCWxncGBt4DUAEGhiMMDHwH9DcwcArAhdgEGFsZhL/LglwHA/P6HBgYgj4dYAAAf14ZMFtvYvoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit( (2 + x) / (4 - x ** 2), x, 2, '+')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAKBAMAAACkp3isAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnaZZhC7ie9EVM0iq905P93KAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAb0lEQVQIHWNgEDJhYGAIMQlgYGBJquhk4NoZWMjAcJiBdf+EtQwM7AwMBQwMEb82MDAwTmBQYGDg/D8BqNwBxGSovwAkFMDMbV8YGHgWMCxkYOBdsH8CA3sAA3MAaxJD3JcYkOm1uQIMDNq/FzAAAO/uGCbPqITdAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit( (2 + x) / (4 - x ** 2), x, 2, '-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.340\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\pm 0} (2 + x ) ^ \\frac{1}{x} }}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAKBAMAAACkp3isAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnaZZhC7ie9EVM0iq905P93KAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAb0lEQVQIHWNgEDJhYGAIMQlgYGBJquhk4NoZWMjAcJiBdf+EtQwM7AwMBQwMEb82MDAwTmBQYGDg/D8BqNwBxGSovwAkFMDMbV8YGHgWMCxkYOBdsH8CA3sAA3MAaxJD3JcYkOm1uQIMDNq/FzAAAO/uGCbPqITdAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit( (2 + x) ** (1 / x), x, 0, '+')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXaZIs1UEN1mu6tEMu+iw/3TAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAV0lEQVQIHWNgYBQyYWBgCGOocGDgbGBgXMDAtYGB7TsDvwEDzzeGeiD5k8FfgIH3E5j8yFAvwMDzCSTL9o2BawED53cGzgcMzAEMDG0MdRcYGAqV9jIAAMSUErnbiJtLAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit( (2 + x) ** (1 / x), x, 0, '-')"
   ]
  }
 ],
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