【微分積分】3-6-3 漸近展開の極限計算｜要点まとめ

\(\displaystyle \sin x=x-\frac{x^3}{6}+o(x^3)\ \ \ (x\to0)\)より、
\(\displaystyle x-\sin x=\frac{x^3}{6}+o(x^3)\ \ \ (x\to0)\)
よって、
\(\displaystyle \lim_{x\to0}\frac{x-\sin x}{x^3}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{x^3}{6}+o(x^3)}{x^3}\)
\(\displaystyle =\lim_{x\to0}\left(\frac{1}{6}+\frac{o(x^3)}{x^3}\right)\)
\(\displaystyle =\frac{1}{6}+0\)
\(\displaystyle =\frac{1}{6}\)

\(\displaystyle e^x=1+x+\frac{x^2}{2}+o(x^2)\ \ \ (x\to0)\)より、
\(\displaystyle e^x-1-x=\frac{x^2}{2}+o(x^2)\ \ \ (x\to0)\)
よって、
\(\displaystyle \lim_{x\to0}\frac{e^x-1-x}{x^2}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{x^2}{2}+o(x^2)}{x^2}\)
\(\displaystyle =\lim_{x\to0}\left(\frac{1}{2}+\frac{o(x^2)}{x^2}\right)\)
\(\displaystyle =\frac{1}{2}+0\)
\(\displaystyle =\frac{1}{2}\)

\(\displaystyle \log(1+x)=x-\frac{x^2}{2}+o(x^2)\ \ \ (x\to0)\)より、
\(\displaystyle \log(1+x)-x=-\frac{x^2}{2}+o(x^2)\ \ \ (x\to0)\)
よって、
\(\displaystyle \lim_{x\to0}\frac{\log(1+x)-x}{x^2}\)
\(\displaystyle =\lim_{x\to0}\frac{-\frac{x^2}{2}+o(x^2)}{x^2}\)
\(\displaystyle =\lim_{x\to0}\left(-\frac{1}{2}+\frac{o(x^2)}{x^2}\right)\)
\(\displaystyle =-\frac{1}{2}+0\)
\(\displaystyle =-\frac{1}{2}\)

\(\displaystyle =\lim_{x\to0}\frac{(\frac{x^2}{2}+o(x^2))(\frac{x^3}{6}+o(x^3))}{x(\frac{x^2}{2}+o(x^2))^2}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{x^5}{12}+o(x^5)}{\frac{x^5}{4}+o(x^5)}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{1}{12}+\frac{o(x^5)}{x^5}}{\frac{1}{4}+\frac{o(x^5)}{x^5}}\)
\(\displaystyle =\frac{\frac{1}{12}+0}{\frac{1}{4}+0}\)
\(\displaystyle =\frac{1}{3}\)

\(\displaystyle =\lim_{x\to0}\frac{(x-\frac{x^3}{6}+o(x^3))-x(1-\frac{x^2}{2}+o(x^3))}{x^2(x+o(x))}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{x^3}{3}+o(x^3)}{x^3+o(x^3)}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{1}{3}+\frac{o(x^3)}{x^3}}{1+\frac{o(x^3)}{x^3}}\)
\(\displaystyle =\frac{\frac{1}{3}+0}{1+0}\)
\(\displaystyle =\frac{1}{3}\)

\(\displaystyle =\lim_{x\to0}\frac{-\frac{2}{3}x^3+o(x^3)}{-\frac{1}{2}x^3+o(x^3)}\)
\(\displaystyle =\lim_{x\to0}\frac{-\frac{2}{3}+\frac{o(x^3)}{x^3}}{-\frac{1}{2}+\frac{o(x^3)}{x^3}}\)
\(\displaystyle =\frac{-\frac{2}{3}+0}{-\frac{1}{2}+0}\)
\(\displaystyle =\frac{4}{3}\)

\(\displaystyle =\lim_{x\to0}\frac{\sin x-x}{x^2\sin x}\)
\(\displaystyle =\lim_{x\to0}\frac{(x-\frac{x^3}{6}+o(x^3))-x}{x^2(x+o(x))}\)
\(\displaystyle =\lim_{x\to0}\frac{-\frac{x^3}{6}+o(x^3)}{x^3+o(x^3)}\)
\(\displaystyle =\lim_{x\to0}\frac{-\frac{1}{6}+\frac{o(x^3)}{x^3}}{1+\frac{o(x^3)}{x^3}}\)
\(\displaystyle =\frac{-\frac{1}{6}+0}{1+0}\)
\(\displaystyle =-\frac{1}{6}\)

\(\displaystyle =\lim_{x\to0}\frac{(1+x^2+o(x^2))-(1-\frac{x^2}{2}+o(x^2))}{x(x+o(x))}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{3}{2}x^2+o(x^2)}{x^2+o(x^2)}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{3}{2}+\frac{o(x^2)}{x^2}}{1+\frac{o(x^2)}{x^2}}\)
\(\displaystyle =\frac{\frac{3}{2}+0}{1+0}\)
\(\displaystyle =\frac{3}{2}\)

\(\displaystyle =\lim_{x\to0}\frac{(x+\frac{x^3}{3}+o(x^3))-x}{x-(x-\frac{x^3}{6}+o(x^3))}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{x^3}{3}+o(x^3)}{\frac{x^3}{6}+o(x^3)}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{1}{3}+\frac{o(x^3)}{x^3}}{\frac{1}{6}+\frac{o(x^3)}{x^3}}\)
\(\displaystyle =\frac{\frac{1}{3}+0}{\frac{1}{6}+0}\)
\(=2\)

\(e^x+\log(1-x)-1\)
\(\displaystyle =\left(1+x+\frac{x^2}{2}+\frac{x^3}{6}+o(x^3)\right)\)
\(\displaystyle \ \ \ \ \ +\left(-x-\frac{x^2}{2}-\frac{x^3}{3}+o(x^3)\right)-1\)
\(\displaystyle =-\frac{x^3}{6}+o(x^3)\)
\(x-\tan^{-1}x\)
\(\displaystyle =x-(x-\frac{x^3}{3}+o(x^3))\)
\(\displaystyle =\frac{x^3}{3}+o(x^3)\)
よって、
\(\displaystyle \lim_{x\to0}\frac{e^x+\log(1-x)-1}{x-\tan^{-1}x}\)
\(\displaystyle =\lim_{x\to0}\frac{-\frac{x^3}{6}+o(x^3)}{\frac{x^3}{3}+o(x^3)}\)
\(\displaystyle =\lim_{x\to0}\frac{-\frac{1}{6}+\frac{o(x^3)}{x^3}}{\frac{1}{3}+\frac{o(x^3)}{x^3}}\)
\(\displaystyle =\frac{-\frac{1}{6}+0}{\frac{1}{3}+0}\)
\(=-\frac{1}{2}\)

\(\sin x+\cos x-\sqrt{1+2x}\)
\(\displaystyle =\left(x-\frac{x^3}{6}+o(x^3)\right)\)
\(\displaystyle \ \ \ \ \ +\left(1-\frac{x^2}{2}+o(x^3)\right)\)
\(\displaystyle \ \ \ \ \ -\left(1+x-\frac{x^2}{2}-\frac{x^3}{2}+o(x^3)\right)\)
\(\displaystyle =-\frac{2}{3}x^3+o(x^3)\)
\(x^2(e^x-1)\)
\(=x^2(x+o(x))\)
\(=x^3+o(x^3)\)
よって、
\(\displaystyle \lim_{x\to0}\frac{\sin x+\cos x-\sqrt{1+2x}}{x^2(e^x-1)}\)
\(\displaystyle =\lim_{x\to0}\frac{-\frac{2}{3}x^3+o(x^3)}{x^3+o(x^3)}\)
\(\displaystyle =\lim_{x\to0}\frac{-\frac{2}{3}+\frac{o(x^3)}{x^3}}{1+\frac{o(x^3)}{x^3}}\)
\(\displaystyle =\frac{-\frac{2}{3}+0}{1+0}\)
\(\displaystyle =-\frac{2}{3}\)

\(\displaystyle =\lim_{x\to0}\frac{e-(e-\frac{e}{2}x+o(x))}{x}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{e}{2}x+o(x)}{x}\)
\(\displaystyle =\lim_{x\to0}\left(\frac{e}{2}+\frac{o(x)}{x}\right)\)
\(\displaystyle =\frac{e}{2}+0\)
\(\displaystyle =\frac{e}{2}\)

\(e^{\sin x}\)
\(\displaystyle =1+\sin x+\frac{\sin^2x}{2}+\frac{\sin^3x}{6}+o(\sin^3x)\)
\(\displaystyle =1+\left(x-\frac{x^3}{6}+o(x^3)\right)\)
\(\displaystyle \ \ \ \ \ +\frac{1}{2}\left(x-\frac{x^3}{6}+o(x^3)\right)^2\)
\(\displaystyle \ \ \ \ \ +\frac{1}{6}\left(x-\frac{x^3}{6}+o(x^3)\right)^3+o(x^3)\)
\(\displaystyle =1+\left(x-\frac{x^3}{6}\right)+\frac{1}{2}x^2+\frac{1}{6}x^3+o(x^3)\)
\(\displaystyle =1+x+\frac{x^2}{2}+o(x^3)\)
\(e^x-e^{\sin x}\)
\(\displaystyle =\left(1+x+\frac{x^2}{2}+\frac{x^3}{6}+o(x^3)\right)\)
\(\displaystyle \ \ \ \ \ -\left(1+x+\frac{x^2}{2}+o(x^3)\right)\)
\(\displaystyle =\frac{x^3}{6}+o(x^3)\)
\(x-\sin x\)
\(\displaystyle =x-\left(x-\frac{x^3}{6}+o(x^3)\right)\)
\(\displaystyle =\frac{x^3}{6}+o(x^3)\)
よって、
\(\displaystyle \lim_{x\to0}\frac{e^x-e^{\sin x}}{x-\sin x}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{x^3}{6}+o(x^3)}{\frac{x^3}{6}+o(x^3)}\)
\(\displaystyle =\lim_{x\to0}\frac{\frac{1}{6}+\frac{o(x^3)}{x^3}}{\frac{1}{6}+\frac{o(x^3)}{x^3}}\)
\(\displaystyle =\frac{\frac{1}{6}+0}{\frac{1}{6}+0}\)
\(=1\)