【微分積分】3-6-3 漸近展開の極限計算｜問題集

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

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