(1)

\begin{align*} \frac{2}{\pi}\lim_{k\rightarrow\infty}\tan^{\bullet}\left(kx\right) & =\frac{2}{\pi}\cdot\begin{cases} \frac{\pi}{2} & 0<x\\ 0 & x=0\\ -\frac{\pi}{2} & x<0 \end{cases}\\ & =\begin{cases} 1 & 0<x\\ 0 & x=0\\ -1 & x<0 \end{cases}\\ & =\sgn\left(x\right) \end{align*}

(2)

\begin{align*} \lim_{k\rightarrow\infty}\tanh\left(kx\right) & =\lim_{k\rightarrow\infty}\frac{e^{kx}-e^{-kx}}{e^{kx}+e^{-kx}}\\ & =\begin{cases} 1 & 0<x\\ 0 & x=0\\ -1 & x<0 \end{cases}\\ & =\sgn\left(x\right) \end{align*}