第14章§6 - 15

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