Evaluate the following limit: $\lim_{x \to 0} (1+x)^{\tan(\frac{1}{x})}$

The limit does not exist.

Note that there are arbitrarily large positive solutions to the equation $\tan \theta=\theta$ (sketch the graphs: tan regularly goes from $0$ to $+\infty$ and there is a solution in each such segment). For every solution, setting $x=1/\theta$ gives $(1+x)^{\tan(1/x)}=(1+1/\theta)^\theta\geq 2$.

This gives an infinite sequence approaching $0$ from above for which the function exceeds $2$; you can easily find a sequence for which the function takes the value $1$ (e.g. $x_n=1/(n\pi)$), so there is no limit.