第14章§6 - 6

$\sin \left(x^2\right),(\sin x{)}^2$のテイラー多項式は $$\begin{array}{rl}\sin \left(x^2\right)& =x^2-\frac{x^6}{3!}-\cdots \\ (\sin x{)}^2& ={(x-\frac{x^3}{3!}+\cdots )}^2=x^2-\frac{2}{3!}{x}^4+\cdots \end{array}$$ となるから $$\begin{array}{rl}\underset{x\to 0}{lim}\frac{\sin \left(x^2\right)}{(\sin x{)}^2}& =\underset{x\to 0}{lim}\frac{x^2-\frac{x^6}{3!}-\cdots }{x^2-\frac{2}{3!}{x}^4+\cdots }\\ & =\underset{x\to 0}{lim}\frac{1-\frac{x^4}{3!}-\cdots }{1-\frac{2}{3!}{x}^2+\cdots }\\ & =1\end{array}$$