mathanalysis_practice105.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   "source": [
    "%%html\n",
    "<style>\n",
    "@import url('https://fonts.googleapis.com/css?family=Roboto&effect=3d|brick-sign|');\n",
    "span {font-family:Roboto; color:black; text-shadow:4px 4px 4px #aaa;}  \n",
    "h1,h2 {font-family:Roboto; color:slategray; text-shadow:4px 4px 4px #aaa;} \n",
    "div.output_subarea, div.output_area pre{font-family:Roboto; font_size:120%; color:darkblue;}\n",
    "</style> "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ссылки / Links\n",
    "\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",
    "[Функция. Предел функции в точке и бесконечности. Теоремы о пределах](https://www.youtube.com/watch?v=wH2kbfndYjM)\n",
    "\n",
    "[Бесконечно малые и бесконечно большие функции и их свойства](https://www.youtube.com/watch?v=1vigfVdPlOE)\n",
    "\n",
    "[Непрерывность функции. Основные элементарные функции. Замечательные пределы.](https://www.youtube.com/watch?v=0J4Wm_4YKdw&t=57s)\n",
    "\n",
    "[Примеры решения задач](http://ru.solverbook.com/primery-reshenij/)\n",
    "\n",
    "[Пределы с корнями: примеры решений](https://xn--24-6kcaa2awqnc8dd.xn--p1ai/predely-s-kornyami.html)\n",
    "\n",
    "---\n",
    "\n",
    "[Limits and continuity](https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new)"
   ]
  },
  {
   "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)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "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": [
    "#### # 5.262\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 1} \\frac{1} {x} = 1}}$\n",
    "\n",
    "$ f(x) = 1/x \\implies D(f) = (-\\infty,0) \\cup (0, +\\infty)$\n",
    "\n",
    "$ \\forall \\varepsilon > 0 \\ \\exists \\delta(\\varepsilon) > 0: 0 < | x - x_0 | < \\delta(\\varepsilon) \\implies | f(x) - a | < \\varepsilon, \\ \\forall x  $\n",
    "\n",
    "$ 0 < | x - 1 | < \\delta(\\varepsilon) \\implies | \\frac{1}{x} - 1 | < \\varepsilon, \\ \\forall x  $\n",
    "\n",
    "$ -\\varepsilon < \\frac{1}{x} -1 < \\varepsilon \\iff  1 - \\varepsilon < \\frac{1}{x}  < 1 + \\varepsilon \\iff \n",
    "\\frac{1}{1 - \\varepsilon} > x > \\frac{1}{1 + \\varepsilon} \\\\\n",
    "\\iff \\frac{1}{1 + \\varepsilon} - 1 < x - 1 < \\frac{1}{1 - \\varepsilon} - 1 \\\\\n",
    "\\iff \\frac{-\\varepsilon}{1 + \\varepsilon} < x - 1 < \\frac{\\varepsilon}{1 - \\varepsilon} \n",
    "\\iff \\frac{-\\varepsilon}{1 - \\varepsilon} < x - 1 < \\frac{\\varepsilon}{1 - \\varepsilon} \\\\\n",
    "\\iff | x - 1 |< \\frac{\\varepsilon}{1 - \\varepsilon} \\implies \\delta(\\varepsilon) = \\frac{\\varepsilon}{1 - \\varepsilon}\n",
    "$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.09090909090909091, -0.11111111111111112, True)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-0.1/(1 + 0.1), -0.1/(1-0.1), -0.1/(1 + 0.1) > -0.1/(1-0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ε = 0.1 ; δ(ε) = 0.111111\n",
      "ε = 0.01 ; δ(ε) = 0.010101\n",
      "ε = 0.001 ; δ(ε) = 0.001001\n"
     ]
    }
   ],
   "source": [
    "e = [0.1, 0.01, 0.001]\n",
    "for el in e:\n",
    "    print ('ε =', el, '; δ(ε) =', round(el/(1-el),6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x): return 1/x\n",
    "x = numpy.arange(0.8, 1.2,0.01)\n",
    "y = [f(e) for e in x]\n",
    "plt.axhline(1+0.1, c='r', ls='--'); plt.axvline(1+0.1/(1-0.1), c='r', ls='--')\n",
    "plt.text(1.01,1.11,' ε', color='r',fontsize=15)\n",
    "plt.text(1+0.1/(1-0.1),1.15,' δ(ε)', color='r',fontsize=15)\n",
    "plt.axhline(1, ls='dotted', label='y = 1', c='b'); plt.axvline(1, ls='dotted', label='x = 1', c='g');\n",
    "plt.axhline(1-0.1, c='r', ls='--'); plt.axvline(1-0.1/(1-0.1), c='r', ls='--')\n",
    "plt.text(1.01,0.87,'-ε', color='r',fontsize=15) \n",
    "plt.text(1-0.1/(1-0.1),1.15,' -δ(ε)', color='r',fontsize=15)\n",
    "plt.plot(x,y,label='y = f(x)'); plt.legend(loc=3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.265\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 1-0} f(x) = - \\infty}}$\n",
    "\n",
    "$ \\forall E > 0 \\ \\exists \\delta > 0 \\ | \\ \\forall x : - \\delta < x - 1 < 0 \\implies f(x) < - E $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.267\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to +\\infty} f(x) = + \\infty}}$\n",
    "\n",
    "$ \\forall E > 0 \\ \\exists \\Delta > 0 \\ | \\ \\forall x : x > \\Delta \\implies f(x) > E $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.269\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} f(x) = 2}}$\n",
    "\n",
    "$ \\forall \\varepsilon > 0 \\ \\exists \\Delta > 0 \\ | \\ \\forall x : |x| > \\Delta \\implies \\mid f(x) - 2 \\mid <  \\varepsilon $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.271\n",
    "$\\large{\\displaystyle{\\lim_{x \\to -\\infty} f(x) = \\infty}}$\n",
    "\n",
    "$ \\forall E > 0 \\ \\exists \\Delta > 0 \\ | \\ \\forall x : - \\Delta < x < 0 \\implies \\mid f(x)\\mid > E $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.272\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\frac{x^2 - 2} {3x^2 - 5x + 1} }} = \\frac {0^2 - 2}{3 * 0^2 - 5 * 0 + 1} = -2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOBAMAAAAoFKpzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZu6uJRO92ZlRfaCDxAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAX0lEQVQIHWNggALWNHUHGJuBIYKB8xeC18jAsBvB2y/A0B8A5zYzMPQ7wHlAhj0yh+UPMo85gYFByBgETIDCachSvEApBHjJwIqwgWUCA48AXC44LQ1oJQzo////iQEAjikQ0wptu4sAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-2$$"
      ],
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import x\n",
    "sympy.limit((x ** 2 - 2) / (3 * x ** 2 - 5 * x + 1), x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAOBAMAAABA5yhLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZu6uJRO92ZlRfaCDxAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvklEQVQYGWNggALWNHUHMHPmsQCYGANDBAPnLxCP6QGjKUK0kYFhN4h3mYFhOkJ0vwBDP0hrBQPDewG4cDMDQ78DkPeNgWH9A7gokGEPxIwg0QtIoix/gBzWnwwM/geQRJkTgByw6AQGIWMQMAEKpAExAyNQ7foDIBYE8CaAaaC57x9AREDkSwZWkMvKGRjuI1zGMoGBB8S7zcCQBFIEAcFpac0M8goMzA8Yt8DEGBj0////xMCcwMA454QDAwCc1SsQiBMAKwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$-2.0$$"
      ],
      "text/plain": [
       "-2.0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x): return (x ** 2 - 2) / (3 * x ** 2 - 5 * x + 1)\n",
    "f(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.280\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\frac{1}{2}} \\frac{8 * x^3 - 1} {6 * x^2 - 5 * x + 1}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOBAMAAADDIxFwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiTIi3USrzZm7VBDvdmaluOHpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAApklEQVQIHS3OQQqCQBTG8b+RpRajHkDwCC3aN8to1RHCE0iraBFBBC3tBl4haCnovjN0h1pEoCi9CWfxmB/fe28GCMMYOVZ0kjpfj7TRnvsMdriZQGmsDK8xAbgbcx1Xf+AfsWv8bXQ2LkUd5YFpLloEDFrKF86j1xtfoz6mM8BuGaaoTiRbvBq3z+RR1WDLXCWZqnBSuFLkky+sKGL57SWBJdyeCT80zyeDxPpBXQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$6.0$$"
      ],
      "text/plain": [
       "6.00000000000000"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "sympy.limit((8 * x ** 3 - 1)/(6 * x ** 2 - 5 * x + 1), x, 0.5 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### $\\frac{(2 * x - 1) * ( 4 * x^2 + 2 * x + 1)} {(2 * x - 1) * ( 3 * x - 1)} = \\frac{( 4 * x^2 + 2 * x + 1)} {( 3 * x - 1)}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOBAMAAADDIxFwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiTIi3USrzZm7VBDvdmaluOHpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAApklEQVQIHS3OQQqCQBTG8b+RpRajHkDwCC3aN8to1RHCE0iraBFBBC3tBl4haCnovjN0h1pEoCi9CWfxmB/fe28GCMMYOVZ0kjpfj7TRnvsMdriZQGmsDK8xAbgbcx1Xf+AfsWv8bXQ2LkUd5YFpLloEDFrKF86j1xtfoz6mM8BuGaaoTiRbvBq3z+RR1WDLXCWZqnBSuFLkky+sKGL57SWBJdyeCT80zyeDxPpBXQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$6.0$$"
      ],
      "text/plain": [
       "6.0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x): return (4 * x ** 2 + 2 * x + 1) / (3 * x - 1)\n",
    "f(1/2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.282\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} (\\frac{x^3 } {2 x^2 - 1} - \\frac{x^2 } {2 x + 1})}}$\n",
    "\n",
    "### $\\frac{x^3 ( 2x + 1) - x^2 (2 x^2 - 1)} {(2 x^2 - 1) ( 2x + 1)} = \\frac{x^3 + x^2} {4 x^3 + 2 x^2 - 2x - 1}$\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} \\frac{x^3 + x^2} {4 x^3 + 2 x^2 - 2x - 1} = \\frac{1}{4}}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAMt0Qq5nNdrvvZokiRASejI8AAAAJcEhZcwAADsQAAA7EAZUrDhsAAABaSURBVBgZY2BgEDJgYGBgds0DUQwMdVSkPOapFIANJYt4BwIPiNQqClGnAqbYIZRUCpi3AUyxHwBTpxjA1AYwxXUATB1nAFNXXFxeeoN1TIEY9gJMhb1rZwAAZVwdNgSBadYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{4}$$"
      ],
      "text/plain": [
       "1/4"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "sympy.limit((x ** 3 / (2 * x ** 2 - 1) - x ** 2 / (2 * x + 1)), x, oo )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.285\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 1} \\bigg(\\frac{x + 2 } {x^2 - 5x + 4} + \\frac{x - 4} {3(x^2 - 3x + 2)}\\bigg)}}$\n",
    "\n",
    "### $\\frac{x + 2 } {x^2 - 5x + 4} + \\frac{x - 4} {3(x^2 - 3x + 2)} = \\frac{x + 2 } {(x - 1)(x - 4)} + \\frac{x - 4} {3(x - 1)(x - 2)} = $\n",
    "### $ = \\frac{3(x + 2)(x - 2)  + (x - 4)^2} {3(x - 1)(x - 2)(x - 4)} = \\frac{3x^2 - 12  + x^2 - 8x + 16} {3(x - 1)(x - 2)(x - 4)} = $\n",
    "### $ = \\frac{4x^2 - 8x + 4} {3(x - 1)(x - 2)(x - 4)} = \\frac{4(x^2 - 2x + 1)} {3(x - 1)(x - 2)(x - 4)} = $\n",
    "### $ = \\frac{4(x - 1)^2} {3(x - 1)(x - 2)(x - 4)} = \\frac{4(x - 1)} {3(x - 2)(x - 4)} $\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 1} \\frac{4(x - 1)} {3(x - 2)(x - 4)} = \\frac{4(1 - 1)} {3(1 - 2)(1 - 4)} = 0}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXaZIs1UEN1mu6tEMu+iw/3TAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAV0lEQVQIHWNgYBQyYWBgCGOocGDgbGBgXMDAtYGB7TsDvwEDzzeGeiD5k8FfgIH3E5j8yFAvwMDzCSTL9o2BawED53cGzgcMzAEMDG0MdRcYGAqV9jIAAMSUErnbiJtLAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "sympy.limit(((x + 2) / (x ** 2 - 5 * x + 4) + (x - 4) / (3 * (x ** 2 - 3 * x + 2))), x, 1 )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.291\n",
    "$\\large{\\displaystyle{\\lim_{h \\to 0} \\frac{\\sqrt{x + h} - \\sqrt{x}}{h} }, x > 0}$\n",
    "\n",
    "### $\\frac{\\sqrt{x + h} - \\sqrt{x}}{h} = \\frac{(\\sqrt{x + h} - \\sqrt{x}) * (\\sqrt{x + h} + \\sqrt{x})}{h * (\\sqrt{x + h} + \\sqrt{x})} = \\frac{x + h - x}{h * (\\sqrt{x + h} + \\sqrt{x})} = \\frac{h }{h * (\\sqrt{x + h} + \\sqrt{x})} = \\frac{1}{\\sqrt{x + h} + \\sqrt{x}}$\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{h \\to 0} \\frac{1}{\\sqrt{x + h} + \\sqrt{x}}} = \\frac{1}{2 \\sqrt{x}} , x > 0}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{0.5}{x^{0.5}}$$"
      ],
      "text/plain": [
       "     -0.5\n",
       "0.5⋅x    "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = symbols('h')\n",
    "sympy.limit(((x + h) ** 0.5 - x ** 0.5) / h, h, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.297\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\frac{\\sqrt{x^2 + 4} - 2}{\\sqrt{x^2 + 9} - 3} }}$\n",
    "\n",
    "### $\\frac{\\sqrt{x^2 + 4} - 2}{\\sqrt{x^2 + 9} - 3} = \\frac{(\\sqrt{x^2 + 4} - 2) (\\sqrt{x^2 + 9} + 3)}{(\\sqrt{x^2 + 9} - 3) (\\sqrt{x^2 + 9} + 3)} = \\frac{(\\sqrt{x^2 + 4} - 2) (\\sqrt{x^2 + 9} + 3)}{x^2 } = $\n",
    "### $= \\frac{(\\sqrt{x^2 + 4} - 2) (\\sqrt{x^2 + 9} + 3) (\\sqrt{x^2 + 4} + 2) }{x^2 (\\sqrt{x^2 + 4} + 2)} = \\frac{x^2 (\\sqrt{x^2 + 9} + 3)}{x^2 (\\sqrt{x^2 + 4} + 2)} = \\frac{ \\sqrt{x^2 + 9} + 3}{\\sqrt{x^2 + 4} + 2}$\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to 0} \\frac{\\sqrt{x^2 + 9} + 3}{\\sqrt{x^2 + 4} + 2} }} = \\frac{3 + 3} {2 + 2} = 1.5$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAPBAMAAADjSHnWAAAALVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAMt0Qq5nNdrvvIolmRIT9yeQAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABqSURBVAgdY2CAAkYGBisgk1EZzOd8lgRkmIQ9BvOY9LQNgAx2CI8FLEYU73ipAEIfhwHXWwQPyGpC4cUtgNvQxsAQB9QItW8SA4MfQuUGBoYZUJ5dAsNSBvbJDAwsnY/cGZgCGLhCXQ0YALyMGbrP4AQEAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$1.5$$"
      ],
      "text/plain": [
       "1.50000000000000"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "sympy.limit(((x ** 2 + 4) ** 0.5 - 2) / ((x ** 2 + 9) ** 0.5 - 3), x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.299\n",
    "$\\large{\\displaystyle{\\lim_{x \\to +\\infty} (\\sqrt{x - a} - \\sqrt{x})}}$\n",
    "\n",
    "### $\\sqrt{x - a} - \\sqrt{x} = \\frac{(\\sqrt{x - a} - \\sqrt{x})(\\sqrt{x - a} + \\sqrt{x})}{\\sqrt{x - a} + \\sqrt{x}} = \\frac{ - a}{\\sqrt{x - a} + \\sqrt{x}}$\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to +\\infty} \\frac{ - a}{\\sqrt{x - a} + \\sqrt{x}}}} = \\frac{-a} {+\\infty} = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXaZIs1UEN1mu6tEMu+iw/3TAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAV0lEQVQIHWNgYBQyYWBgCGOocGDgbGBgXMDAtYGB7TsDvwEDzzeGeiD5k8FfgIH3E5j8yFAvwMDzCSTL9o2BawED53cGzgcMzAEMDG0MdRcYGAqV9jIAAMSUErnbiJtLAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,a = symbols('x a')\n",
    "sympy.limit(- a / ((x ** 2 - a) ** 0.5 + x ** 0.5 ), x, +oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.301\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} (\\sqrt{4x^2 - 7x + 4} - 2x)}}$\n",
    "\n",
    "### $ \\sqrt{4x^2 - 7x + 4} - 2x = \\frac{(\\sqrt{4x^2 - 7x + 4} - 2x)(\\sqrt{4x^2 - 7x + 4} + 2x)}{\\sqrt{4x^2 - 7x + 4} + 2x} = $\n",
    "### $ = \\frac{4x^2 - 7x + 4 - 4x^2}{\\sqrt{4x^2 - 7x + 4} + 2x} = \\frac{- 7x + 4}{2x(\\sqrt{1 - 7/4x + 1/x^2} + 1)} = $\n",
    "### $ = \\frac{- 7}{2(\\sqrt{1 - 7/4x + 1/x^2} + 1)} + \\frac{2}{x(\\sqrt{1 - 7/4x + 1/x^2} + 1)}$\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} \\bigg(\\frac{- 7}{2(\\sqrt{1 - 7/4x + 1/x^2} + 1)} + \\frac{2}{x(\\sqrt{1 - 7/4x + 1/x^2} + 1)}\\bigg)}} = \\frac{-7}{2  * (1 + 1)} + 0 = -1.75$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADMAAAAPBAMAAAC7N+rPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMquZdrvvImZUiURkEzbAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvklEQVQYGXWMvwqCcBRGD5hYQiBBS4tBTxC1Rj6Ck20lBa1tzVFTW9HS2CRFD9DqGxi9SP/2zH5qiuU33Ms9h/tBJkctAAWopXihKQ7P9/1V6dEIL4Gqw7vYF2s8cqVu5/McRwmVDQ5yDMMdKRfMHAVlI1ftQT5NjVRnVAh9KGrq649SbwIuoNL6pA3xV7ku1GQnlhixkmYwh4nxq3QTNjBIzLfwasMZthml18ELuIOyTpS8fPaQbDi4oI4tjTeNGC10l6j0ggAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$-1.75$$"
      ],
      "text/plain": [
       "-1.75000000000000"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "sympy.limit(((4 * x ** 2 - 7 * x + 4) ** 0.5 - 2 * x) , x, oo)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### # 5.302\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} x ^ {\\frac{3}{2}} (\\sqrt{x^3 + 2} - \\sqrt{x^3 - 2})}}$\n",
    "\n",
    "### $x ^ {\\frac{3}{2}} (\\sqrt{x^3 + 2} - \\sqrt{x^3 - 2}) = x ^ {\\frac{3}{2}} \\frac{(\\sqrt{x^3 + 2} - \\sqrt{x^3 - 2})(\\sqrt{x^3 + 2} + \\sqrt{x^3 - 2})}{\\sqrt{x^3 + 2} + \\sqrt{x^3 - 2}} = $\n",
    "\n",
    "### $ = \\frac{4x ^ {\\frac{3}{2}} }{\\sqrt{x^3 + 2} + \\sqrt{x^3 - 2}} = \\frac{4x ^ {\\frac{3}{2}} }{x ^ {\\frac{3}{2}}(\\sqrt{1 + 2/x^3} + \\sqrt{1 - 2/x^3})} = \\frac{4 }{\\sqrt{1 + 2/x^3} + \\sqrt{1 - 2/x^3}}$\n",
    "\n",
    "$\\large{\\displaystyle{\\lim_{x \\to \\infty} \\frac{4 }{\\sqrt{1 + 2/x^3} + \\sqrt{1 - 2/x^3}}}} = \\frac{4}{\\sqrt{1 + 0} + \\sqrt{1-0}} = \\frac{4}{2} = 2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOBAMAAAAoFKpzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNuzKriUQQ7912ZlSugimNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAApElEQVQIHTWNIQ7CUAyGvxeykJdBgsJglscFEHgMQe8GaDKxCRIMd9i7AQqDIZyABYFGg+A55BIQIEhGK2jS9v/S9i/GjVM0Vqce9Gl/FFrBHmABO6UbLGHvyWWFGdSeDHI9fEIZpHOUtEoXEdFXinnDvBIRJ38qRDhJrMzKCrqJkrrUAR4Y/TCFuycq6HihKwxh4FzGZkQc7BYmTfNSW7s+p/wAf9godrK6dw4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2.0$$"
      ],
      "text/plain": [
       "2.00000000000000"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols('x')\n",
    "sympy.limit(x ** 1.5 * ((x ** 3 + 2) ** 0.5 - (x ** 3 - 2) ** 0.5) , x, oo)"
   ]
  }
 ],
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