Answer
$\ln 3$
Work Step by Step
Consider: $\lim\limits_{\theta \to 0}f(x)=\lim\limits_{\theta \to 0} \dfrac{3^{ \sin\theta}-1}{\theta}$
We need to check that the limit has an indeterminate form.
Thus, $f(0)=\dfrac{0}{0}$
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
$\lim\limits_{\theta \to 0}\dfrac{(3^{ \sin\theta})(\ln 3) (\cos \theta)}{1}=(3^{ \sin (0)})(\ln 3) (\cos (0))=\ln 3$
