Answer
$$0$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{x \to \infty } \frac{{\ln {x^{100}}}}{{\sqrt x }} \cr
& {\text{Evaluate }} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{{\ln {x^{100}}}}{{\sqrt x }} = \mathop {\lim }\limits_{x \to \infty } \frac{{\ln {{\left( \infty \right)}^{100}}}}{{\sqrt \infty }} = \frac{\infty }{\infty } \cr
& {\text{Apply the LHopital's rule}} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{{\ln {x^{100}}}}{{\sqrt x }} = \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{d}{{dx}}\left[ {\ln {x^{100}}} \right]}}{{\frac{d}{{dx}}\left[ {\sqrt x } \right]}} = \mathop {\lim }\limits_{x \to \infty } \frac{{\frac{{100}}{x}}}{{\frac{1}{{2\sqrt x }}}} \cr
& = \mathop {\lim }\limits_{x \to \infty } \frac{{200\sqrt x }}{x} = \mathop {\lim }\limits_{x \to \infty } \frac{{200}}{{{x^{1/2}}}} \cr
& {\text{Evaluate }} \cr
& \mathop {\lim }\limits_{x \to \infty } \frac{{200}}{{{x^{1/2}}}} = \frac{{200}}{{{{\left( \infty \right)}^{1/2}}}} = 0 \cr} $$
