calculus - How do I take the limit as $n$ goes to $infty$ of $frac{sqrt{n}}{log(n)}$?

How do take this limit:

$$\lim_{n\to\infty} \frac{\sqrt{n}}{\log(n)}$$

I have a feeling that it is infinity, but I'm not sure how to prove it. Should I use L'Hopitals Rule?

Answer

Let $n = e^x$. Note that as $n \rightarrow \infty$, we also have $x \rightarrow \infty$. Hence, $$\lim_{n \rightarrow \infty} \frac{\sqrt{n}}{\log(n)} = \lim_{x \rightarrow \infty} \frac{\exp(x/2)}{x}$$
Note that $\displaystyle \exp(y) > \frac{y^2}{2}$, $\forall y > 0$ (Why?). Hence, we have that $$\lim_{n \rightarrow \infty} \frac{\sqrt{n}}{\log(n)} = \lim_{x \rightarrow \infty} \frac{\exp(x/2)}{x} \geq \lim_{x \rightarrow \infty} \frac{\frac{x^2}{8}}{x} = \lim_{x \rightarrow \infty} \frac{x}{8} = \infty$$