Evaluate a given limit Evaluate the following limit:
$$\lim_{x \rightarrow 4} \left( \frac{1}{13 - 3x} \right) ^ {\tan \frac{\pi x}{8}}$$
I haven't managed to get anything meaningful yet.
Thank you in advance!

Compute the limit of the logarithm:
$$
\lim_{x\to4}\tan\frac{\pi x}{8}\log\frac{1}{13-3x}=
\lim_{x\to4}\frac{-\log(13-3x)}{\cot\frac{\pi x}{8}}
$$
This screams l’Hôpital!
$$
\lim_{x\to4}\frac{\frac{3}{13-3x}}{\frac{\pi}{8}(-1-\cot^2\frac{\pi x}{8})}=\frac{3}{-\pi/8}=-\frac{24}{\pi}
$$
Thus your limit is $e^{-24/\pi}$.
