Answer
$f(x)$ is continuous at $0$.
Work Step by Step
When $f(0)=\lim\limits_{x\to 0} f(x)$, then $f(x)$ will be continuous at $x=0$.
We have:
$\lim\limits_{x\to 0^{-}}f(x)=\lim\limits_{x\to 0^{-}} 2e^x =2e^{0}=2*1=2$
and $\lim\limits_{x\to 0^{+}}f(x)=\lim\limits_{x\to 0^{+}}\dfrac{x^2 (x+2)}{x^2} =2+0=2$
Thus: $\lim\limits_{x\to 0}f(x)=2$
Since, $f(0)=2$, we can see that our result satisfies the statement because $2 = 2$. Therefore, $f(x)$ is continuous at $0$.
