mathanalysis_standardcalc110-115.ipynb

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  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math, sympy, numpy\n",
    "from sympy import *\n",
    "sympy.init_printing(use_unicode=True)\n",
    "import pylab as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[GeoGebra](https://www.geogebra.org/graphing?lang=en)\n",
    "\n",
    "[LaTeX/Mathematics](https://en.wikibooks.org/wiki/LaTeX/Mathematics)\n",
    "\n",
    "[Math24.ru. Ряды Тейлора и Маклорена](http://www.math24.ru/%D1%80%D1%8F%D0%B4%D1%8B-%D1%82%D0%B5%D0%B9%D0%BB%D0%BE%D1%80%D0%B0-%D0%B8-%D0%BC%D0%B0%D0%BA%D0%BB%D0%BE%D1%80%D0%B5%D0%BD%D0%B0.html)\n",
    "\n",
    "[Math24.ru. Бесконечно малые функции](http://www.math24.ru/%D0%B1%D0%B5%D1%81%D0%BA%D0%BE%D0%BD%D0%B5%D1%87%D0%BD%D0%BE-%D0%BC%D0%B0%D0%BB%D1%8B%D0%B5-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8.html)\n",
    "\n",
    "[Math24.ru. Точки разрыва функции](http://www.math24.ru/%D1%82%D0%BE%D1%87%D0%BA%D0%B8-%D1%80%D0%B0%D0%B7%D1%80%D1%8B%D0%B2%D0%B0-%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8.html)\n",
    "\n",
    "[ SymPy Modules Reference. Calculus](https://docs.sympy.org/latest/modules/calculus/index.html)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Вариант 22\n",
    "# Пределы"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Типовой расчет №10\n",
    "\n",
    "Вычислить предел функции $\\large{\\displaystyle{\\lim_{x \\to 0} \\frac { e^x + e^{-x}  - 2 }{\\sin ^ 2 x} }}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение \n",
    "\n",
    "## I Способ\n",
    "\n",
    "Числитель и знаменатель дроби стремятся к 0 при $x \\to 0$, \n",
    "\n",
    "значит дробь является неопределенным выражением вида $\\big[\\frac{0}{0}\\big]$ при $x \\to 0$.\n",
    "\n",
    "Для раскрытия неопределенности преобразуем функцию, используя разложение в ряды Маклорена:\n",
    "\n",
    "$\\large{\\displaystyle{ e^x = 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\overline{o_1(x^4)} }}$\n",
    "\n",
    "$\\large{\\displaystyle{e^{-x} = 1 - x + \\frac{x^2}{2!} - \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\overline{o_2(x^4)} }}$\n",
    "\n",
    "$\\large{\\displaystyle{ \\sin^2 x = \\frac{1}{2}(1 - \\cos 2x) = \\frac{1}{2}(1 - (1 - \\frac{4x^2}{2!} +  \\frac{16x^4}{4!})) + \\overline{o_3(x^4) }= x^2 - \\frac{x^4}{3} + \\overline{o_3(x^4)} }}$\n",
    "\n",
    "$\\overline{o_1(x^4)}, \\ \\overline{o_2(x^4)}, \\ \\overline{o_3(x^4)} \\to 0$ при $x \\to 0$.\n",
    "\n",
    "$\\large{\\displaystyle{ \\frac { e^x + e^{-x}  - 2 }{sin ^ 2 x} = }}$\n",
    "\n",
    "$\\large{\\displaystyle{=  \\frac { 1 + x + \\frac{x^2}{2!} + \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\overline{o_1(x^4)} + 1 - x + \\frac{x^2}{2!} - \\frac{x^3}{3!} + \\frac{x^4}{4!} + \\overline{o_2(x^4)} - 2 }{x^2 - \\frac{x^4}{3} + \\overline{o_3(x^4)}} = }}$\n",
    "\n",
    "$\\large{\\displaystyle{=  \\frac {(x^2 + \\frac {x^4}{12} + \\overline{o_1(x^4)} + \\overline{o_2(x^4)}) / x^2}{(x^2 - \\frac{x^4}{3} + \\overline{o_3(x^4)}) / x^2} = }}$\n",
    "\n",
    "$\\large{\\displaystyle{= \\frac {1 + \\frac {x^2}{12} + \\frac{\\overline{o_1(x^4)}}{x^2} + \\frac{\\overline{o_2(x^4)}}{x^2}} {1 - \\frac{x^2}{3} + \\frac{\\overline{o_3(x^4)}}{x^2}} }}$\n",
    "\n",
    "Учитывая то, что дроби $\\large{\\displaystyle{\\frac {x^2}{12}, \\frac{x^2}{3}, \\frac{\\overline{o_1(x^4)}}{x^2}, \\frac{\\overline{o_2(x^4)}}{x^2}, \\frac{\\overline{o_3(x^4)}}{x^2} }}$\n",
    "\n",
    "являются бесконечно малыми величинами при $x \\to 0$:\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\frac {1 + \\frac {x^2}{12} + \\frac{\\overline{o_1(x^4)}}{x^2} + \\frac{\\overline{o_2(x^4)}}{x^2}} {1 - \\frac{x^2}{3} + \\frac{\\overline{o_3(x^4)}}{x^2}} = \\frac{1 + 0 + 0 + 0}{1 - 0 + 0} = 1}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## II Способ\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Числитель и знаменатель дроби стремятся к 0 при $x \\to 0$, \n",
    "\n",
    "значит дробь является неопределенным выражением вида $\\big[\\frac{0}{0}\\big]$ при $x \\to 0$.\n",
    "\n",
    "Используя арифметические свойства пределов и свойства тригонометрических и показательных функций, \n",
    "\n",
    "выполним элементарные преобразования:\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\frac{ e^x + e^{-x}  - 2 }{\\sin ^ 2 x} = a^2 * \\lim_{x \\to 0} \\frac {e^ x - 2 e^{\\frac{x}{2}} e^{-\\frac{x}{2}} + e^{- x}}{\\sin ^ 2 x} = }}$\n",
    "\n",
    "$\\large{\\displaystyle{ = \\lim_{x \\to 0} \\frac {(e^{\\frac{x}{2}})^2 - 2 e^{\\frac{x}{2}} e^{-\\frac{x}{2}} + (e^{-\\frac{x}{2}})^2}{\\sin ^ 2 x}  = \\lim_{x \\to 0} \\Bigg(\\frac { e^{\\frac{x}{2}} - e^{- \\frac{x}{2}} }{\\sin x}\\Bigg)^2 = }}$\n",
    "\n",
    "$\\large{\\displaystyle{ = \\lim_{x \\to 0} \\Bigg(\\frac {e^{\\frac{x}{2}} - 1 - (e^{- \\frac{x}{2}} - 1)}{\\sin x}\\Bigg)^2 = \\lim_{x \\to 0} \\Bigg(\\frac {e^{\\frac{x}{2}} - 1 - (e^{- \\frac{x}{2}} - 1)}{\\sin \\frac{x}{2}} * \\frac{\\sin \\frac{x}{2}}{\\sin x}\\Bigg)^2 = }}$\n",
    "\n",
    "$\\large{\\displaystyle{ = \\Bigg(\\lim_{x \\to 0} \\frac {e^{\\frac{x}{2}} - 1 - (e^{- \\frac{x}{2}} - 1)}{\\sin \\frac{x}{2}}\\Bigg)^2 * \\Bigg(\\lim_{x \\to 0} \\frac{\\sin \\frac{x}{2}}{\\sin x} \\Bigg)^2 = }}$\n",
    "\n",
    "$\\large{\\displaystyle{ = \\Bigg(\\lim_{x \\to 0} \\frac {e^{\\frac{x}{2}} - 1}{\\sin \\frac{x}{2}} + \\lim_{x \\to 0} \\frac{e^{- \\frac{x}{2}} - 1}{\\sin(-\\frac{x}{2})}\\Bigg)^2 * \\Bigg(\\lim_{x \\to 0} \\frac{\\sin \\frac{x}{2}}{\\sin x} \\Bigg)^2 }} (*)$\n",
    "\n",
    "Очевидно, что $\\frac{x}{2} \\to 0, -\\frac{x}{2} \\to 0$ при $x \\to 0$.\n",
    "\n",
    "Применим асимптотические равенства для бесконечно малых функций:\n",
    "\n",
    "$\\large{\\displaystyle{ (e^t - 1) \\sim t , \\sin t \\sim t \\iff \\frac{e^t - 1}{\\sin t} \\sim 1}}$\n",
    "\n",
    "$\\large{\\displaystyle{ t = x \\implies \\lim_{x \\to 0} \\frac {\\sin x}{x} = 1, \\lim_{x \\to 0} \\frac {x}{\\sin x} = 1}}$\n",
    "\n",
    "$\\large{\\displaystyle{ t = \\frac {x}{2} \\implies \\lim_{x \\to 0} \\frac {e^{\\frac{x}{2}} - 1}{\\sin \\frac{x}{2}} = 1, \\ \\lim_{x \\to 0} \\frac {\\sin \\frac{x}{2}}{\\frac{x}{2}} = 1}}$\n",
    "\n",
    "$\\large{\\displaystyle{ t = -\\frac {x}{2} \\implies \\lim_{x \\to 0} \\frac {e^{-\\frac{x}{2}} - 1}{\\sin(-\\frac{x}{2})} = 1 }}$\n",
    "\n",
    "$\\large{\\displaystyle{ \\Bigg( \\lim_{x \\to 0} \\frac{ \\sin \\frac{x}{2} } {\\sin x}\\Bigg)^2 = \\frac{1}{4} * \\Bigg( \\lim_{x \\to 0} \\frac{ \\sin \\frac{x}{2} }{\\sin x} * \\frac{x}{\\frac{x}{2}} \\Bigg)^2 = }}$\n",
    "\n",
    "$\\large{\\displaystyle{ = \\frac{1}{4} * \\Bigg( \\lim_{x \\to 0} \\frac{ \\sin \\frac{x}{2} }{\\frac{x}{2}} \\Bigg)^2 * \\Bigg( \\lim_{x \\to 0} \\frac{ x }{\\sin x} \\Bigg)^2 = \\frac{1}{4} * 1^2 * 1^2 = \\frac{1}{4} }}$\n",
    "\n",
    "$\\large{\\displaystyle{ \\implies (*) = (1 + 1)^2 * \\frac{1}{4} = 1}}$"
   ]
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    {
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      "text/latex": [
       "$$2.71828182845905$$"
      ],
      "text/plain": [
       "2.71828182845905"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(E)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOBAMAAADgeEClAAAAIVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADdcGRXAAAACnRSTlMAMt0Qq5nNdrvvxbMB0AAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABtJREFUCB1jYGBUZmAwCVvMwMDAThbB0rnIHQBAqQoWQ8kakgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit((E ** x + E ** (-x) - 2) / (sin(x)) ** 2, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Типовой расчет №11\n",
    "\n",
    "Вычислить предел функции $\\large{\\displaystyle{\\lim_{x \\to \\pi} \\frac {\\ln \\cos 2x}{(x - \\pi)^2} }}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение \n",
    "\n",
    "Числитель и знаменатель дроби стремятся к 0 при $x \\to \\pi$, \n",
    "\n",
    "значит дробь является неопределенным выражением вида $\\big[\\frac{0}{0}\\big]$ при $x \\to \\pi$.\n",
    "\n",
    "Для раскрытия неопределенности преобразуем функцию, используя формулы эквивалентности функций.\n",
    "\n",
    "Прежде всего произведем замену переменной $x - \\pi = t \\iff x = t + \\pi; \\ x \\to \\pi \\implies t \\to 0$.\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\pi} \\frac {\\ln \\cos 2x}{(x - \\pi)^2} = \\lim_{t \\to 0} \\frac {\\ln \\cos (2t + 2\\pi)}{t^2} = \\lim_{t \\to 0} \\frac {\\ln (1 + (\\cos 2t - 1))}{t^2} = \\lim_{t \\to 0} \\frac {\\cos 2t - 1}{t^2}}}$\n",
    "\n",
    "Здесь использована формула эквивалентности $\\ln(1 + a) \\sim a$ при $a \\to 0$. \n",
    "\n",
    "Поскольку $cos2t - 1 \\to 0$ при $t \\to 0$, то  $\\ln (1 + (\\cos 2t - 1)) \\sim cos2t - 1$ при $t \\to 0$.\n",
    "\n",
    "Применим формулу синуса половинного угла $\\sin^2 a = \\frac{1 - cos 2 a}{2}$ и арифметические свойства пределов:\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{t \\to 0} \\frac {\\cos 2t - 1}{t^2} = \\lim_{t \\to 0} \\frac {-2\\sin^2t}{t^2} = -2 \\Bigg(\\lim_{t \\to 0} \\frac {\\sin t}{t}\\Bigg)^2}}$\n",
    "\n",
    "С учетом первого замечательного предела $\\displaystyle{\\lim_{a \\to 0} \\frac{\\sin a}{a} = 1}$:\n",
    "\n",
    "$\\large{\\displaystyle{-2 \\Bigg(\\lim_{t \\to 0} \\frac {\\sin t}{t}\\Bigg)^2 = -2 * 1^2 = -2}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
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    {
     "data": {
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      "text/latex": [
       "$$3.14159265358979$$"
      ],
      "text/plain": [
       "3.14159265358979"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOBAMAAAAoFKpzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZu6uJRO92ZlRfaCDxAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAX0lEQVQIHWNggALWNHUHGJuBIYKB8xeC18jAsBvB2y/A0B8A5zYzMPQ7wHlAhj0yh+UPMo85gYFByBgETIDCachSvEApBHjJwIqwgWUCA48AXC44LQ1oJQzo////iQEAjikQ0wptu4sAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-2$$"
      ],
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit(ln(cos(2 * x)) / (x - pi) ** 2, x, pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Типовой расчет №12\n",
    "\n",
    "Вычислить предел функции $\\large{\\displaystyle{\\lim_{x \\to 0} \\Bigg(\\frac {2x + 9}{x + 9}\\Bigg)^{1/(\\sqrt[3]{x + 8} - 2)} }}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение \n",
    "\n",
    "$\\large{\\displaystyle{\\frac {2x + 9}{x + 9} \\to 1; \\frac{1}{\\sqrt[3]{x + 8} - 2} \\to \\infty}}$ при $x \\to 0$, значит\n",
    "\n",
    "функция является неопределенным выражением вида $\\big[1^{\\infty}\\big]$ при $x \\to 0$.\n",
    "\n",
    "Для раскрытия неопределенности воспользуемся \n",
    "\n",
    "- вторым замечательным пределом и \n",
    "- известной схемой его применения.\n",
    "\n",
    "Пусть $g(x) \\to \\infty, h(x) \\to 0$ при $x \\to 0$.\n",
    "\n",
    "Поскольку показательная функция $e^t, \\ \\forall t \\in \\mathbb{R}$ непрерывна, то справедливы следующие равенства:\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0}\\bigg(1 + h(x)\\bigg)^{g(x)} =  \\lim_{x \\to 0}\\Bigg[\\bigg(1 + h(x)\\bigg)^{\\frac{1}{h(x)}}\\Bigg]^{h(x)g(x)} =  \\lim_{x \\to 0} e^{h(x)g(x)} }}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Примения эту схему к данному примеру, получим:\n",
    "    \n",
    "$\\large{\\displaystyle{h(x) = 1 - \\frac {2x + 9}{x + 9} = \\frac {x + 9 - 2x - 9}{x + 9} = -\\frac{x}{x+9} \\to 0}}$ при $x \\to 0$\n",
    "\n",
    "$\\large{\\displaystyle{g(x) = \\frac{1}{\\sqrt[3]{x + 8} - 2} \\to \\infty}}$ при $x \\to 0$\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\Bigg(\\frac {2x + 9}{x + 9}\\Bigg)^{1/(\\sqrt[3]{x + 8} - 2)} =  \\lim_{x \\to 0} e^ {-\\frac{x}{x+9} \\frac{1}{\\sqrt[3]{x + 8} - 2}} = \\lim_{x \\to 0} e^ \\frac{x}{(x + 9)(2 - \\sqrt[3]{x + 8})} }}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Применим тождественные преобразования выражений и арифметические свойства предела:\n",
    "    \n",
    " $\\large{\\displaystyle{ \\frac{x}{(x + 9)(2 - \\sqrt[3]{x + 8})} = \\frac{x}{2x + 18 - x\\sqrt[3]{x + 8} - 9\\sqrt[3]{x + 8}} = }}$\n",
    " \n",
    " $\\large{\\displaystyle{= \\frac{x}{2x + 18 - \\sqrt[3]{x^4 + 8x^3} - \\sqrt[3]{729x + 5832}} }}$\n",
    " \n",
    "$\\large{\\displaystyle{ \\lim_{x \\to 0}\\frac{x}{2x + 18 - \\sqrt[3]{x^4 + 8x^3} - \\sqrt[3]{729x + 5832}} = \\frac{0}{0 + 18 - 0 - \\sqrt[3]{5832}} = 0 \\implies }}$\n",
    "\n",
    "$\\large{\\displaystyle{ \\lim_{x \\to 0} e^ {\\frac{x}{(x + 9)(2 - \\sqrt[3]{x + 8})}} = e^0 = 1}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOBAMAAADgeEClAAAAIVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADdcGRXAAAACnRSTlMAMt0Qq5nNdrvvxbMB0AAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABtJREFUCB1jYGBUZmAwCVvMwMDAThbB0rnIHQBAqQoWQ8kakgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit(((2*x + 9) / (x + 9)) ** ((x + 8) ** (1/3) - 2), x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Типовой расчет №13\n",
    "\n",
    "Вычислить предел функции $\\large{\\displaystyle{\\lim_{x \\to \\infty} \\Bigg(\\frac {x}{x - 1}\\Bigg)^{3 - 2x} }}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение \n",
    "\n",
    "$\\frac {x}{x - 1} \\to 1$; $3 - 2x \\to \\infty$ при $x \\to \\infty$, значит\n",
    "\n",
    "функция является неопределенным выражением вида $\\big[1^{\\infty}\\big]$ при $x \\to \\infty$.\n",
    "\n",
    "Заменим переменную: $ y = \\frac{1}{x} \\iff x = \\frac{1}{y}$, причем $y \\to 0, \\frac{1}{y} \\to \\infty$ при $x \\to \\infty$.\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} \\Bigg(\\frac {x}{x - 1}\\Bigg)^{3 - 2x} = \\lim_{x \\to \\infty} \\Bigg(\\frac {x - 1}{x}\\Bigg)^{2x - 3} = \\lim_{x \\to \\infty} \\Bigg(1 - \\frac {1}{x}\\Bigg)^{2x - 3} = }}$\n",
    "\n",
    "$\\large{\\displaystyle{= \\lim_{y \\to 0} \\bigg(1 - y\\bigg)^{\\frac{2}{y} - 3}  = \\lim_{y \\to 0} \\bigg(1 - y\\bigg)^{\\frac{2 - 3y}{y}} = \\lim_{y \\to 0} \\Bigg[ \\bigg(1 + (-y) \\bigg)^{\\frac{1}{-y}} \\Bigg] ^{3y - 2} (*) }}$\n",
    "\n",
    "Применяя формулу второго замечательного предела, получим:\n",
    "\n",
    "$\\large{\\displaystyle{-y \\to 0, \\frac{1}{-y} \\to \\infty \\implies \\lim_{y \\to 0} \\bigg(1 + (-y) \\bigg)^{\\frac{1}{-y}} = e \\implies}}$.\n",
    "\n",
    "Поскольку показательная функция $e^t, \\ \\forall t \\in \\mathbb{R}$ непрерывна:\n",
    "\n",
    "$\\large{\\displaystyle{ (*) = \\lim_{y \\to 0} e^{3y -2} = e^{-2} }}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAATBAMAAACaQstxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mUTdMiJmu6tUze/kkN0jAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfUlEQVQYGWNgQAJC2huReECmA4MbqsBXhv4CFJGJ6AIMDGooChgY2BOhAuyrgGAFA0OXAKoKjg3sDSgiXeUlqEre//8PVMCobIIqHFbAroCskeMnA3MAsgDXZyNVZD4D/wYULgMDvwKaABtQ4ACyGOcFBpEGZAEGI2MDZD4AXRAW/EmqlZoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$e^{-2}$$"
      ],
      "text/plain": [
       " -2\n",
       "ℯ  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit(( x / (x - 1) ) ** (3 - 2 * x), x, oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Типовой расчет №14\n",
    "\n",
    "Сравнить бесконечно малые при $x \\to x_0$ функции $\\alpha(x)$ и $\\beta(x)$.\n",
    "\n",
    "$\\alpha(x) = \\sin (3x - 2)^2; \\ \\beta(x) = ch(2 - 3x) - 1; \\ x_0 = \\frac{2}{3}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение \n",
    "\n",
    "Чтобы сравнить заданные функции $\\alpha(x)$ и $\\beta(x)$ при $x \\to x_0$, \n",
    "\n",
    "найдем предел частного $\\large{\\displaystyle{\\lim_{x \\to \\frac {2}{3}} \\frac {\\sin (3x - 2)^2}{ch(2 - 3x) - 1} (*)}}$ \n",
    "\n",
    "Заменим переменную: $ y = 2 - 3x \\iff x = \\frac{2 - y}{3}$, причем $y \\to 0, \\frac{1}{y} \\to \\infty$ при $x \\to \\frac {2}{3}$.\n",
    "\n",
    "и подставим соотвествующее выражение для гиперболического косинуса $ch \\ t = (e^t + e^{-t})/2$.\n",
    "\n",
    "$\\large{\\displaystyle{(*) = \\lim_{y \\to 0} \\frac {\\sin (-y)^2}{ch \\ y - 1} = \\lim_{y \\to 0} \\frac {\\sin (y^2)}{(e^y + e^{-y})/2 - 1} = \\lim_{y \\to 0} \\frac {2\\sin (y^2)}{e^y + e^{-y} - 2} (**) }}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Используем правило Лопиталя\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\frac { f(x) }{ g(x) } = \\lim_{x \\to 0} \\frac { f'(x) }{ g'(x) } }}$\n",
    "\n",
    "и дважды продифференцируем числитель и знаменатель:\n",
    "                                                                                             \n",
    "$\\large{\\displaystyle{(**) =  \\lim_{y \\to 0} \\frac {(2\\sin (y^2))'}{(e^y + e^{-y} - 2)'} =  \\lim_{y \\to 0} \\frac {4y\\cos (y^2)}{e^y - e^{-y}} = }}$                                                   \n",
    "\n",
    "$\\large{\\displaystyle{=  \\lim_{y \\to 0} \\frac {(4y\\cos (y^2))'}{(e^y - e^{-y})'} = \\lim_{y \\to 0} \\frac {-8 y^2 sin(y^2) + 4\\cos (y^2)}{e^y + e^{-y}} = \\frac{0 + 4}{1 + 1} = 2}}$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$4 y \\cos{\\left (y^{2} \\right )}$$"
      ],
      "text/plain": [
       "       ⎛ 2⎞\n",
       "4⋅y⋅cos⎝y ⎠"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = symbols('y')\n",
    "diff(2 * sin(y ** 2) , y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{y} - e^{- y}$$"
      ],
      "text/plain": [
       " y    -y\n",
       "ℯ  - ℯ  "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(E ** y + E ** (-y) - 2, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 8 y^{2} \\sin{\\left (y^{2} \\right )} + 4 \\cos{\\left (y^{2} \\right )}$$"
      ],
      "text/plain": [
       "     2    ⎛ 2⎞        ⎛ 2⎞\n",
       "- 8⋅y ⋅sin⎝y ⎠ + 4⋅cos⎝y ⎠"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(diff(2 * sin(y ** 2) , y),y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$e^{y} + e^{- y}$$"
      ],
      "text/plain": [
       " y    -y\n",
       "ℯ  + ℯ  "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(diff(E ** y + E ** (-y) - 2, y), y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNuzKriUQQ7912ZlSugimNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAT0lEQVQIHWNgVDIJYGAQY2D/wcCQycCwioFh/QSGfAGGbAaGfKAEA8N+EMH6B0RyK4BIJRDBqwAinzMwCjCwFjDwTGCQVlICarH///8zAwAXpw31OpgbMwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2$$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "limit(sin((3 * x - 2) ** 2) / (cosh(2 - 3 * x) - 1), x, 2/3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Типовой расчет №15\n",
    "\n",
    "Найти точки разрыва функции и установить их характер.\n",
    "\n",
    "$\\large{\\displaystyle{ f(x) = \\frac{1 - \\cos 6x}{4 x^2} }}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Решение \n",
    "\n",
    "Функция $f(x)$ определена и непрерывна для всех $x \\in \\mathbb{R}$, кроме $x = 0$, т.е.\n",
    "\n",
    "$f(x)$ непрерывна на всей области определения $(-\\infty, 0) \\cup (0, +\\infty)$.\n",
    "\n",
    "В точке $x = 0$ функция $f(x)$ имеет разрыв, т.к. она определена в некоторой проколотой окрестности\n",
    "\n",
    "этой точки, а в самой точке не определена.\n",
    "\n",
    "Для классификации тип разрыва, вычислим односторонние пределы при $x \\to 0$.\n",
    "\n",
    "Учитывая вид неопределенности, при вычислении предела используем правило Лопиталя\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\frac { f(x) }{ g(x) } = \\lim_{x \\to 0} \\frac { f'(x) }{ g'(x) } }}$\n",
    "\n",
    "и дважды продифференцируем числитель и знаменатель:\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0 - 0} \\frac{1 - \\cos 6x}{4 x^2} = \\Bigg[ \\frac{0}{0} \\Bigg] = \\lim_{x \\to 0 - 0} \\frac{(1 - \\cos 6x)'}{(4 x^2)'} = }}$\n",
    "\n",
    "$\\large{\\displaystyle{= \\lim_{x \\to 0 - 0} \\frac{6 \\sin 6x}{8x} = \\lim_{x \\to 0 - 0} \\frac{(6 \\sin 6x)'}{(8x)'} = \\lim_{x \\to 0 - 0} \\frac{36 \\cos 6x}{8} = \\frac {36 * 1}{8} = 4.5}}$.\n",
    "\n",
    "Аналогично\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0 + 0} \\frac{1 - \\cos 6x}{4 x^2} = \\Bigg[ \\frac{0}{0} \\Bigg] = \\lim_{x \\to 0 + 0} \\frac{36 \\cos 6x}{8} = \\frac {36 * 1}{8} = 4.5}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Левосторонний и правосторонний пределы существуют и конечны, значит\n",
    "\n",
    "$x = 0$ является ***точкой разрыва первого рода***.\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0 - 0} f(x) = \\lim_{x \\to 0 + 0} f(x) = 4.5}}$\n",
    "                      \n",
    "Левосторонний и правосторонний пределы равны, значит \n",
    "\n",
    "$x = 0$ является ***точкой устранимого разрыва***."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = numpy.linspace(-2.0,2.0)\n",
    "y = (1 - numpy.cos(6 * x)) / (4 * x ** 2)\n",
    "\n",
    "plt.figure(figsize=(10,5)); plt.grid(c='steelblue',ls='--',lw=0.5)\n",
    "plt.plot(x, y, c='steelblue', lw=2, label=r'$f (x) = \\frac{1 - \\cos 6x}{4 x^2}$')\n",
    "plt.axvline(0, c='darkred',ls='--',lw=2)\n",
    "\n",
    "plt.xlabel(r'$x$', fontsize=15); plt.ylabel(r'$y$', fontsize=15)\n",
    "plt.legend(fontsize=15, shadow=True);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( 6 \\sin{\\left (6 x \\right )}, \\quad 36 \\cos{\\left (6 x \\right )}\\right )$$"
      ],
      "text/plain": [
       "(6⋅sin(6⋅x), 36⋅cos(6⋅x))"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "diff(1 - cos(6 * x) , x), diff(diff(1 - cos(6 * x) , x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( 8 x, \\quad 8\\right )$$"
      ],
      "text/plain": [
       "(8⋅x, 8)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(4 * x ** 2, x), diff(diff(4 * x ** 2, x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAmasQ3XYy7yK7zYlEZlRwlz41AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqklEQVQYGWNgYBRUZmBgYHZlSG9gYGAqYOAUYGDgX8DA/YGBwX4CA+s3CO8PRO4vUGUNQ9I/oA7D6KdAOSBg/AymuCYwMPAVM/QnMDCwOjPMBYnt3ngARJEJ/oMA0BVEgEapaUBVHAoM+w8A3WnAwF7AwMD+mYHpB5DnwMAEcScLxKz8AKBSBgZHMMkoAKYugkm+DQwdQEY2A0Mw0DBHQSEHBgYuoBsWMAAAzM0orm8wOggAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{9}{2}$$"
      ],
      "text/plain": [
       "9/2"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit( (1 - cos(6 * x)) / (4 * x ** 2), x, 0, '+')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAmasQ3XYy7yK7zYlEZlRwlz41AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqklEQVQYGWNgYBRUZmBgYHZlSG9gYGAqYOAUYGDgX8DA/YGBwX4CA+s3CO8PRO4vUGUNQ9I/oA7D6KdAOSBg/AymuCYwMPAVM/QnMDCwOjPMBYnt3ngARJEJ/oMA0BVEgEapaUBVHAoM+w8A3WnAwF7AwMD+mYHpB5DnwMAEcScLxKz8AKBSBgZHMMkoAKYugkm+DQwdQEY2A0Mw0DBHQSEHBgYuoBsWMAAAzM0orm8wOggAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{9}{2}$$"
      ],
      "text/plain": [
       "9/2"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit( (1 - cos(6 * x)) / (4 * x ** 2), x, 0, '-')"
   ]
  }
 ],
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