Answer
$$\lim\limits _{x \rightarrow 0} \tan \left(\frac{\pi}{4} \cos \left(\sin x^{1 / 3}\right)\right)=\frac{1}{\sqrt 2}
$$
The function is continuous at $ x=0$.
Work Step by Step
Given $$\lim\limits _{x \rightarrow 0} \tan \left(\frac{\pi}{4} \cos \left(\sin x^{1 / 3}\right)\right)
$$
So,
\begin{aligned}a) L&=\lim\limits _{x \rightarrow 0} \tan \left(\frac{\pi}{4} \cos \left(\sin x^{1 / 3}\right)\right)\\
&= \tan \left(\frac{\pi}{4} \cos \left(\sin 0\right)\right)\\
&= \tan \left(\frac{\pi}{4} \cos0\right)\\
&= \tan \left(\frac{\pi}{4} \right)\\
&=\frac{1}{\sqrt 2}\\
\end{aligned}
Since $$ f(x)=\tan \left(\frac{\pi}{4} \cos \left(\sin x^{1 / 3}\right)\right)$$
\begin{aligned}b) f(0)&= \tan \left(\frac{\pi}{4} \cos \left(\sin 0\right)\right)\\
&= \tan \left(\frac{\pi}{4} \cos0\right)\\
&= \tan \left(\frac{\pi}{4} \right)\\
&=\frac{1}{\sqrt 2}\\
\end{aligned}
From (a), (b) since $\lim \limits_{x \rightarrow 0} f(x)=f(0)=\frac{1}{\sqrt 2},$ the function is continuous at $ x=0$.
