Question

Evaluate the limit $\displaystyle{\lim\limits_{\theta \to 0} \dfrac{\sin{(\theta^2)}}{\theta}}$ using the l'Hopital's Rule.

Answer 1

Since we have an indeterminate form of type $\frac{0}{0}$, we can apply the l'Hopital's rule.

$\lim_{\theta \to 0} \frac{\sin{\theta^2}}{\theta}=\lim_{\theta \to 0} \frac{\frac{d}{d\theta}(\sin{\theta^2})}{\frac{d}{d\theta}(\theta)}=\lim_{\theta \to 0}2 \theta cos{\theta^2}=0\\\\ \text{Hence }\\ \lim_{\theta \to 0} \frac{\sin{\theta^2}}{\theta}=0$