数学 - Python - JavaScript - 解析学 - 関数の近似、極限の計算 - 極限の計算(不定形の極限、ロピタルの定理(L'Hospital))

解析入門〈2〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門〈2〉(松坂 和夫(著)、岩波書店)の第6章(関数の近似、極限の計算)、6.2(極限の計算)、問題1、2.を取り組んでみる。

1. 1. 
      $\begin{array}{l}x=\frac{1}{y}\\ x\to \infty \Rightarrow y\to 0\\ \underset{y\to 0}{\lim }\frac{e^y-1}{y}\\ =\underset{y\to 0}{\lim }{e}^y\\ =1\end{array}$
   2. 
      $\begin{array}{l}f\left(x\right)=x^{\frac{1}{x}}\\ \log f\left(x\right)=\frac{\log x}{x}\\ \underset{x\to \infty }{\lim }\log f\left(x\right)\\ =\underset{x\to \infty }{\lim }\frac{\log x}{x}\\ =\underset{x\to \infty }{\lim }\frac{\frac{1}{x}}{1}\\ =0\\ \underset{x\to \infty }{\lim }{x}^{\frac{1}{x}}=1\end{array}$
   3. 
      $\begin{array}{l}\underset{x\to 0}{\lim }\frac{e^{x\log a}-e^{x\log b}}{x}\\ =\underset{x\to 0}{\lim }\left(e^{x\log a}\log a-e^{x\log b}\log b\right)\\ =\log a-\log b\end{array}$
   4. 
      $\begin{array}{l}\underset{x\to 0}{\lim }\frac{1-\frac{1}{\sqrt{1-x^2}}}{3x^2}\\ =\underset{x\to 0}{\lim }\frac{\frac{\frac{1}{2}{\left(1-x^2\right)}^{-\frac{1}{2}}\left(-2x\right)}{1-x^2}}{6x}\\ =\underset{x\to 0}{\lim }\frac{-{\left(1-x^2\right)}^{-\frac{1}{2}}}{6}\\ =-\frac{1}{6}\end{array}$
   5. 
      $\begin{array}{l}\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin x-1}{\cos x}\\ =\underset{x\to \frac{\pi }{2}}{\lim }\frac{\cos x}{-\sin x}\\ =0\end{array}$
   6. 
      $\begin{array}{l}\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin x+x\cos x}{-\sin x}\\ =-1\end{array}$
2. 1. 
      $\begin{array}{l}\underset{x\to 0}{\lim }\frac{1-\frac{1}{1+x}}{2x}\\ =\underset{x\to 0}{\lim }\frac{\frac{1}{{\left(1+x\right)}^2}}{2}\\ =\frac{1}{2}\end{array}$
   2. 
      $\begin{array}{l}\underset{x\to 0}{\lim }\frac{x^2-{\sin }^{2}x}{x^2{\sin }^{2}x}\\ =\underset{x\to 0}{\lim }\frac{x^2-{\sin }^{2}x}{x^4}·\frac{x^2}{{\sin }^{2}x}\\ =\underset{x\to 0}{\lim }\frac{x^2-{\sin }^{2}x}{x^4}\\ =\underset{x\to 0}{\lim }\frac{2x-2\sin x\cos x}{4x^3}\\ =\underset{x\to 0}{\lim }\frac{2x-\sin 2x}{4x^3}\\ =\underset{x\to 0}{\lim }\frac{2-2\cos 2x}{12x^2}\\ =\underset{x\to 0}{\lim }\frac{1-\cos 2x}{6x^2}\\ =\underset{x\to 0}{\lim }\frac{2\sin 2x}{12x}\\ =\underset{x\to 0}{\lim }\frac{4\cos 2x}{12}\\ =\frac{1}{3}\end{array}$
   3. 
      $\begin{array}{l}\underset{x\to 0}{\lim }\frac{1-\cos x}{\frac{1}{{\cos }^{2}x}-1}\\ =\underset{x\to 0}{\lim }\frac{{\cos }^{2}x-{\cos }^{3}x}{1-{\cos }^{2}x}\\ =\underset{x\to 0}{\lim }\frac{-2\cos x\sin x+3\cos x\sin x}{2\cos x\sin x}\\ =\underset{x\to 0}{\lim }\frac{-2+3}{2}\\ =\frac{1}{2}\end{array}$
   4. 
      $\begin{array}{l}\frac{x}{\log x}=\frac{1}{\frac{1}{x}\log x}=\frac{1}{\log x^{\frac{1}{x}}}\\ y=x^{\frac{1}{x}}\\ \log y=\frac{\log x}{x}\\ x\to 0\Rightarrow y\to 1\\ \underset{y\to 1}{\lim }\frac{y-1}{\log y}\\ =\underset{y\to 1}{\lim }\frac{1}{\frac{1}{y}}\\ =1\end{array}$
   5. 
      $\begin{array}{l}y=\frac{1}{\sin x}\\ x\to 0+\Rightarrow y\to \infty \\ xy=\frac{x}{\sin x}\\ f\left(x\right)=y^x\\ \log f\left(x\right)=x\log y\\ \log f\left(x\right)=xy\frac{\log y}{y}\\ x\to 0+\Rightarrow xy\to 1\\ y\to \infty \Rightarrow \frac{\log y}{y}\to 0\\ \underset{x\to 0+}{\lim }f\left(x\right)=1\\ \underset{x\to 0+}{\lim }{\left(\frac{1}{\sin x}\right)}^x=1\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Limit, pi, E, plot

x = symbols('x')
expr = (E - (1 + x) ** (1 / x)) / x
for dir in ['+', '-']:
    l = Limit(expr, x, 0, dir=dir)
    pprint(l)
    pprint(l.doit())

p = plot(expr, show=False)
p.save('sample1.svg')

入出力結果(Terminal, IPython)

$ ./sample1.py
       x _______    
     - ╲╱ x + 1  + ℯ
 lim ───────────────
x─→0⁺       x       
ℯ
─
2
       x _______    
     - ╲╱ x + 1  + ℯ
 lim ───────────────
x─→0⁻       x       
ℯ
─
2
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample1.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let f = (x) => (1 + x) ** (1 / x),
    g = (x) => Math.E - f(x),
    h = (x) => x,
    l = (x) => g(x) / h(x);

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [];

    [[f, 'red'], [g, 'green'], [h, 'blue'], [l, 'brown']].forEach((o) => {
        let [fn, color] = o;
        
        for (let x = x1; x <= x2; x += dx) {
            let y = fn(x);
            
            if (Math.abs(y) < Infinity) {
                points.push([x, y, color]);
            }
        }
        
    });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);

    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =