Chapter 4 - Section 4.5 - Indeterminate Forms and L'Hôpital's Rule - Exercises - Page 248: 21

$2$

Consider: $\lim\limits_{x \to 0}f(x)=\lim\limits_{x \to 0}\dfrac{x^2}{\ln (\sec x)}$ We need to check that the limit has an indeterminate form. Thus, $f(0)=\dfrac{0^2}{\ln (\sec 0)}=\dfrac{0}{0}$ The limit shows an indeterminate form. Thus, apply L-Hospital's rule: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$ Then $\lim\limits_{x \to 0}\dfrac{2}{\sec^2 x}=\dfrac{2}{1}=2$