Little-o notation for $\ln(x)$

Question

I want to Show the following: $\ln(x) = o (e^{\sqrt{\ln x}})$ (Little-o Notation). So I need to Show that: $\displaystyle{\lim_{x \rightarrow \infty} \frac{\ln(x)}{e^{\sqrt{\ln x}}}=0}.$

Can you help me, please?

Answer 1

HINT

Let $x=e^{y^2} \to \infty,\, y\to \infty\,$ therefore

$$\frac{\ln(x)}{e^{\sqrt{\ln x}}}=\frac{y^2}{e^{y}}$$

Answer 2

As gimusi noted, you want to prove $\lim_{y\to\infty}y^2\exp -y=0$. The function is continuous and non-negative for $y\ge 0$ with only one turning point, and famously integrates to the finite value $2$, which implies the limit.