Finding a limit using High School methods

One of my students approached me with a question about a limit. (She doesn't know the idea of a limit.) I was able to find the limit quite easily by using L'Hopital's Rule. However, they don't know about L'Hopital's Rule, and so a proof cannot use it. The limit is as follows:

$$\lim_{x \to \infty} \frac{\ln^2x + 2\ln x}{x}=0$$

I must be missing a simple trick. The proof cannot use L'Hopital's Rule, nor can it be a formal $\varepsilon\delta$-proof. Neither of these ideas have been introduced in the course. We have done Taylor Series.

Can anyone see a simple way, of finding this limit?

Answer

Let $x=e^u$. We are looking at $\frac{u^2+2u}{e^u}$.

For the $\frac{u^2}{e^u}$ part, use the fact that for positive $u$, we have $e^u\gt \frac{u^3}{3!}$. (This comes from the Taylor series.)

The $\frac{2u}{e^u}$ part is done in the same way.