Answer
$1$
Work Step by Step
Our aim is to evaluate the limit for $\lim\limits_{x \to \infty} [\ln (x)]^{1/x}$
Let us consider that $y=\lim\limits_{x \to \infty} [\ln (x)]^{1/x} \implies \ln y=\dfrac{\ln [\ln (x)]}{x}$
Apply L-Hospital's rule which can be defined as: $\lim\limits_{x \to a} \dfrac{P(x) }{Q(x)}=\lim\limits_{x \to a}\dfrac{P'(x)}{Q'(x)}$
where, $a$ can be any real number, infinity or negative infinity.
$\lim\limits_{x \to \infty} \ln y=\lim\limits_{x \to \infty} \dfrac{(1/\ln x) (1/x)}{1}\\=\dfrac{1}{x \times \ln x}\\=0$
Thus, we have: $\ln y=0 \implies y=e^0=1$
