第14章§6 - 16

$\log (1+x^2),(\sin x{)}^2$のテイラー多項式は $$\begin{array}{rl}\log (1+x^2)& =x^2-\frac{x^4}{2}+\cdots \\ (\sin x{)}^2& =x^2-\frac{2}{3!}{x}^4+\cdots \end{array}$$ となるから $$\begin{array}{rl}\underset{x\to 0}{lim}\frac{\log (1+x^2)}{(\sin x{)}^2}& =\underset{x\to 0}{lim}\frac{x^2-\frac{x^4}{2}+\cdots }{x^2-\frac{2}{3!}{x}^4+\cdots }\\ & =\underset{x\to 0}{lim}\frac{1-\frac{x^2}{2}+\cdots }{1-\frac{2}{3!}{x}^2+\cdots }\\ & =1\end{array}$$