Answer
$1$
Work Step by Step
L'Hospital's rule states that $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{A'(x)}{B'(x)}$
Here, $\ln f(x)=\frac{1}{x}\ln (\ln x) \implies f(x)=e^{( \frac{\ln(\ln x)}{x})}$
Now, $e^{\lim\limits_{x \to \infty} ( \frac{\ln(\ln x)}{x})}=\dfrac{\infty}{\infty}$
This shows an indeterminate form of the limit, so we need to use L'Hospital's rule.
$e^{\lim\limits_{x \to \infty} ( \frac{1/x \ln x}{1})}=e^{0}=1$