{
 "metadata": {
  "name": "limit_examples_advanced"
 }, 
 "name": "limit_examples_advanced", 
 "nbformat": 2, 
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code", 
     "collapsed": true, 
     "input": "from sympy import *", 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 1
    }, 
    {
     "cell_type": "code", 
     "collapsed": true, 
     "input": "x = Symbol(\"x\")", 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 2
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(exp(x)*exp(x**2)*(erf(x+1/exp(x))-erf(x)), x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 3, 
       "text": "2/sqrt(pi)"
      }
     ], 
     "prompt_number": 3
    }, 
    {
     "cell_type": "markdown", 
     "source": "The examples here show the limit computation on exp-log expressions (from Gruntz' thesis pp. 122 to 123)"
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.1"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(x)*(exp(1/x-exp(-x))-exp(1/x))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 4, 
       "text": "(-exp(1/x) + exp(-exp(-x) + 1/x))*exp(x)"
      }
     ], 
     "prompt_number": 4
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 5, 
       "text": "-1"
      }
     ], 
     "prompt_number": 5
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.2"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(x)*(exp(1/x+exp(-x)+exp(-x**2)) - exp(1/x-exp(-exp(x))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 6, 
       "text": "(-exp(-exp(-exp(x)) + 1/x) + exp(exp(-x**2) + exp(-x) + 1/x))*exp(x)"
      }
     ], 
     "prompt_number": 6
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 7, 
       "text": "1"
      }
     ], 
     "prompt_number": 7
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.3"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(x-exp(-x))/(1-1/x)) - exp(exp(x))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 8, 
       "text": "exp(exp(x - exp(-x))/(1 - 1/x)) - exp(exp(x))"
      }
     ], 
     "prompt_number": 8
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 9, 
       "text": "oo"
      }
     ], 
     "prompt_number": 9
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.4"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(exp(x)/(1-1/x))) - exp(exp(exp(x)/(1-1/x-log(x)**(-log(x)))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 10, 
       "text": "exp(exp(exp(x)/(1 - 1/x))) - exp(exp(exp(x)/(1 - log(x)**(-log(x)) - 1/x)))"
      }
     ], 
     "prompt_number": 10
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 11, 
       "text": "-oo"
      }
     ], 
     "prompt_number": 11
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.5"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(exp(x+exp(-x)))) / exp(exp(exp(x)))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 12, 
       "text": "exp(-exp(exp(x)))*exp(exp(exp(x + exp(-x))))"
      }
     ], 
     "prompt_number": 12
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 13, 
       "text": "oo"
      }
     ], 
     "prompt_number": 13
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.6"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(exp(x))) / exp(exp(exp(x-exp(-exp(x)))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 14, 
       "text": "exp(exp(exp(x)))*exp(-exp(exp(x - exp(-exp(x)))))"
      }
     ], 
     "prompt_number": 14
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 15, 
       "text": "oo"
      }
     ], 
     "prompt_number": 15
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.7"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(exp(x))) / exp(exp(exp(x-exp(-exp(exp(x))))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 16, 
       "text": "exp(exp(exp(x)))*exp(-exp(exp(x - exp(-exp(exp(x))))))"
      }
     ], 
     "prompt_number": 16
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 17, 
       "text": "1"
      }
     ], 
     "prompt_number": 17
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.8"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(x)) / exp(exp(x-exp(-exp(exp(x)))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 18, 
       "text": "exp(exp(x))*exp(-exp(x - exp(-exp(exp(x)))))"
      }
     ], 
     "prompt_number": 18
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 19, 
       "text": "1"
      }
     ], 
     "prompt_number": 19
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.9"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "log(x)**2 * exp(sqrt(log(x))*(log(log(x)))**2 * exp(sqrt(log(log(x))) * (log(log(log(x))))**3)) / sqrt(x)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 20, 
       "text": "exp(exp(sqrt(log(log(x)))*log(log(log(x)))**3)*sqrt(log(x))*log(log(x))**2)*log(x)**2/sqrt(x)"
      }
     ], 
     "prompt_number": 20
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 21, 
       "text": "0"
      }
     ], 
     "prompt_number": 21
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.10"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "(x*log(x)*(log(x*exp(x)-x**2))**2) / (log(log(x**2+2*exp(exp(3*x**3*log(x))))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 22, 
       "text": "x*log(x)*log(-x**2 + x*exp(x))**2/log(log(x**2 + 2*exp(exp(3*x**3*log(x)))))"
      }
     ], 
     "prompt_number": 22
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 23, 
       "text": "1/3"
      }
     ], 
     "prompt_number": 23
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.11"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "(exp(x*exp(-x)/(exp(-x)+exp(-2*x**2/(x+1)))) - exp(x))/x", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 24, 
       "text": "(-exp(x) + exp(x*exp(-x)/(exp(-2*x**2/(x + 1)) + exp(-x))))/x"
      }
     ], 
     "prompt_number": 24
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 25, 
       "text": "-exp(2)"
      }
     ], 
     "prompt_number": 25
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.12"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "(3**x + 5**x)**(1/x)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 26, 
       "text": "(3**x + 5**x)**(1/x)"
      }
     ], 
     "prompt_number": 26
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 27, 
       "text": "5"
      }
     ], 
     "prompt_number": 27
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.13"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "x/log(x**(log(x**(log(2)/log(x)))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 28, 
       "text": "x/log(x**log(x**(log(2)/log(x))))"
      }
     ], 
     "prompt_number": 28
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 29, 
       "text": "oo"
      }
     ], 
     "prompt_number": 29
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.14"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(2*log(x**5+x)*log(log(x)))) / exp(exp(10*log(x)*log(log(x))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 30, 
       "text": "exp(-exp(10*log(x)*log(log(x))))*exp(exp(2*log(x**5 + x)*log(log(x))))"
      }
     ], 
     "prompt_number": 30
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 31, 
       "text": "oo"
      }
     ], 
     "prompt_number": 31
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.15"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "4*exp(exp(S(5)/2*x**(-S(5)/7)+ S(21)/8*x**(S(6)/11)+2*x**(-8)+S(54)/17*x**(S(49)/45) ))**8 / (9*log(log(-log(S(4)/3*x**(-S(5)/14))))**(S(7)/6))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 32, 
       "text": "4*exp(8*exp(54*x**(49/45)/17 + 21*x**(6/11)/8 + 2/x**8 + 5/(2*x**(5/7))))/(9*log(log(-log(4/(3*x**(5/14)))))**(7/6))"
      }
     ], 
     "prompt_number": 32
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 33, 
       "text": "oo"
      }
     ], 
     "prompt_number": 33
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.16"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "(exp(4*x*exp(-x)/(1/exp(x)+1/exp(2*x**2/(x+1)))) - exp(x)) / exp(x)**4", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 34, 
       "text": "(-exp(x) + exp(4*x*exp(-x)/(exp(-2*x**2/(x + 1)) + exp(-x))))*exp(-4*x)"
      }
     ], 
     "prompt_number": 34
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 35, 
       "text": "1"
      }
     ], 
     "prompt_number": 35
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.17"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(x*exp(-x)/(exp(-x)+exp(-2*x**2/(x+1))))/exp(x)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 36, 
       "text": "exp(-x)*exp(x*exp(-x)/(exp(-2*x**2/(x + 1)) + exp(-x)))"
      }
     ], 
     "prompt_number": 36
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 37, 
       "text": "1"
      }
     ], 
     "prompt_number": 37
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.18"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "(exp(exp(-x/(1+exp(-x))))*exp(-x/(1+exp(-x/(1+exp(-x)))))*exp(exp(-x+exp(-x/(1+exp(-x)))))) / (exp(-x/(1+exp(-x))))**2 - exp(x) + x", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 38, 
       "text": "x - exp(x) + exp(2*x/(1 + exp(-x)))*exp(-x/(1 + exp(-x/(1 + exp(-x)))))*exp(exp(-x/(1 + exp(-x))))*exp(exp(-x + exp(-x/(1 + exp(-x)))))"
      }
     ], 
     "prompt_number": 38
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 39, 
       "text": "2"
      }
     ], 
     "prompt_number": 39
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.19"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "log(x)*(log(log(x)+log(log(x))) - log(log(x))) / (log(log(x)+log(log(log(x)))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 40, 
       "text": "(log(log(x) + log(log(x))) - log(log(x)))*log(x)/log(log(x) + log(log(log(x))))"
      }
     ], 
     "prompt_number": 40
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 41, 
       "text": "1"
      }
     ], 
     "prompt_number": 41
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.20"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp((log(log(x+exp(log(x)*log(log(x)))))) / (log(log(log(exp(x)+x+log(x))))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 42, 
       "text": "exp(log(log(x + exp(log(x)*log(log(x)))))/log(log(log(x + exp(x) + log(x)))))"
      }
     ], 
     "prompt_number": 42
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 43, 
       "text": "E"
      }
     ], 
     "prompt_number": 43
    }, 
    {
     "cell_type": "markdown", 
     "source": "The following examples show limit computation on special functions (from Gruntz' thesis p. 126)"
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.21"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(x)*(sin(1/x+exp(-x))-sin(1/x+exp(-x**2)))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 44, 
       "text": "(sin(exp(-x) + 1/x) - sin(exp(-x**2) + 1/x))*exp(x)"
      }
     ], 
     "prompt_number": 44
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 45, 
       "text": "1"
      }
     ], 
     "prompt_number": 45
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.22"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(x)) * (exp(sin(1/x+exp(-exp(x)))) - exp(sin(1/x)))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 46, 
       "text": "(-exp(sin(1/x)) + exp(sin(exp(-exp(x)) + 1/x)))*exp(exp(x))"
      }
     ], 
     "prompt_number": 46
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 47, 
       "text": "1"
      }
     ], 
     "prompt_number": 47
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.23"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "(erf(x-exp(-exp(x))) - erf(x)) * exp(exp(x)) * exp(x**2)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 48, 
       "text": "(-erf(x) + erf(x - exp(-exp(x))))*exp(x**2)*exp(exp(x))"
      }
     ], 
     "prompt_number": 48
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 49, 
       "text": "-2/sqrt(pi)"
      }
     ], 
     "prompt_number": 49
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.24"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "(Ei(x-exp(-exp(x))) - Ei(x)) *exp(-x)*exp(exp(x))*x", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 50, 
       "text": "x*(-Ei(x) + Ei(x - exp(-exp(x))))*exp(-x)*exp(exp(x))"
      }
     ], 
     "prompt_number": 50
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 51, 
       "text": "-1"
      }
     ], 
     "prompt_number": 51
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.25"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp((log(2)+1)*x) * (zeta(x+exp(-x)) - zeta(x))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 52, 
       "text": "(-zeta(x) + zeta(x + exp(-x)))*exp(x*(log(2) + 1))"
      }
     ], 
     "prompt_number": 52
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "#limit(_, x, oo)", 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 53
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.26"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(x)*(gamma(x+exp(-x)) - gamma(x))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 54, 
       "text": "(-gamma(x) + gamma(x + exp(-x)))*exp(x)"
      }
     ], 
     "prompt_number": 54
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 55, 
       "text": "oo"
      }
     ], 
     "prompt_number": 55
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.27"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(gamma(x-exp(-x))*exp(1/x)) - exp(gamma(x))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 56, 
       "text": "exp(exp(1/x)*gamma(x - exp(-x))) - exp(gamma(x))"
      }
     ], 
     "prompt_number": 56
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "#limit(_, x, oo)", 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 57
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.28"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "(gamma(x+1/gamma(x)) - gamma(x)) / log(x)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 58, 
       "text": "(-gamma(x) + gamma(x + 1/gamma(x)))/log(x)"
      }
     ], 
     "prompt_number": 58
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 59, 
       "text": "1"
      }
     ], 
     "prompt_number": 59
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.29"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "x * (gamma(x-1/gamma(x)) - gamma(x) + log(x))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 60, 
       "text": "x*(log(x) - gamma(x) + gamma(x - 1/gamma(x)))"
      }
     ], 
     "prompt_number": 60
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 61, 
       "text": "1/2"
      }
     ], 
     "prompt_number": 61
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.30"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "((gamma(x+1/gamma(x)) - gamma(x))/log(x) - cos(1/x))*x*log(x)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 62, 
       "text": "x*((-gamma(x) + gamma(x + 1/gamma(x)))/log(x) - cos(1/x))*log(x)"
      }
     ], 
     "prompt_number": 62
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 63, 
       "text": "-1/2"
      }
     ], 
     "prompt_number": 63
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.31"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "gamma(x+1)/sqrt(2*pi) - exp(-x)*(x**(x+S(1)/2) + x**(x-S(1)/2)/12)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 64, 
       "text": "-(x**(x - 1/2)/12 + x**(x + 1/2))*exp(-x) + sqrt(2)*gamma(x + 1)/(2*sqrt(pi))"
      }
     ], 
     "prompt_number": 64
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 65, 
       "text": "oo"
      }
     ], 
     "prompt_number": 65
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.32"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "log(gamma(gamma(x)))/exp(x)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 66, 
       "text": "exp(-x)*log(gamma(gamma(x)))"
      }
     ], 
     "prompt_number": 66
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 67, 
       "text": "oo"
      }
     ], 
     "prompt_number": 67
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.33"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(digamma(digamma(x))))/x", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 68, 
       "text": "exp(exp(polygamma(0, polygamma(0, x))))/x"
      }
     ], 
     "prompt_number": 68
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 69, 
       "text": "exp(-1/2)"
      }
     ], 
     "prompt_number": 69
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.34"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(digamma(log(x))))/x", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 70, 
       "text": "exp(exp(polygamma(0, log(x))))/x"
      }
     ], 
     "prompt_number": 70
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 71, 
       "text": "exp(-1/2)"
      }
     ], 
     "prompt_number": 71
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.35"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "exp(exp(exp(digamma(digamma(digamma(x))))))/x", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 72, 
       "text": "exp(exp(exp(polygamma(0, polygamma(0, polygamma(0, x))))))/x"
      }
     ], 
     "prompt_number": 72
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 73, 
       "text": "0"
      }
     ], 
     "prompt_number": 73
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.36"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "besselj(2,x)*exp(x*(2*log(2+sqrt(3))-sqrt(3)))*sqrt(x)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 74, 
       "text": "sqrt(x)*exp(x*(-sqrt(3) + 2*log(sqrt(3) + 2)))*besselj(2, x)"
      }
     ], 
     "prompt_number": 74
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "#limit(_, x, oo)", 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 75
    }, 
    {
     "cell_type": "markdown", 
     "source": "Eqn 8.37"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "Max(x, exp(x))/log(Min(exp(-x), exp(-exp(x))))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 76, 
       "text": "Max(x, exp(x))/log(Min(exp(-x), exp(-exp(x))))"
      }
     ], 
     "prompt_number": 76
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "#limit(_, x, oo)", 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 77
    }, 
    {
     "cell_type": "markdown", 
     "source": "Some other examples"
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "digamma(digamma(digamma(x)))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 78, 
       "text": "polygamma(0, polygamma(0, polygamma(0, x)))"
      }
     ], 
     "prompt_number": 78
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 79, 
       "text": "oo"
      }
     ], 
     "prompt_number": 79
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "loggamma(loggamma(x))", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 80, 
       "text": "loggamma(loggamma(x))"
      }
     ], 
     "prompt_number": 80
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": "limit(_, x, oo)", 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 81, 
       "text": "oo"
      }
     ], 
     "prompt_number": 81
    }
   ]
  }
 ]
}