The value of k for which $f(x)=\left\{\begin{aligned} x\left(1+x e^{-1 / x^2} \sin \frac{1}{x^4}\right)^{e^{1 / x^2}} & , x \neq 0 \\ k \quad \quad ~~~~& , x=0 \end{aligned}\right.$ is continuous at x = 0, is __________.

For f(x) to be continuous at x = 0, we must have

$\lim\limits_{x \rightarrow 0} f(x)=f(0)$

$\Rightarrow \lim\limits_{x \rightarrow 0}\left(1+x e^{-1 / x^2} \sin \frac{1}{x^4}\right)^{e^{1 / x^2}}=k$

$\Rightarrow e^{\lim\limits_{x \rightarrow 0}\left(x e^{1 / x^2} e^{-1 / x^2} \sin \frac{1}{x^4}\right)}=k$

$\Rightarrow e^{\lim\limits_{x \rightarrow 0} x \sin \frac{1}{x^4}}=k$

$\Rightarrow e^0=k \Rightarrow k=1$