Answer
$1$
Work Step by Step
Here, we have $\ln f(x)=\frac{1}{x}\ln (\ln x)$
and $f(x)=e^{( \frac{\ln(\ln x)}{x})}$
Now, $e^{\lim\limits_{x \to \infty} ( \frac{\ln(\ln x)}{x})}=\dfrac{\infty}{\infty}$
This shows an indeterminate form of limit, thus we will apply L-Hospital's rule such as:
$\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{p'(x)}{q'(x)}$
$e^{\lim\limits_{x \to \infty} ( \frac{1/x \ln x}{1})}=e^{( \frac{1}{\infty})}$
or, $e^{0}=1$
