Finding limit of the function by power series estimation

Question

I want to prove that the limit of function $\displaystyle \lim_{x \to \infty}\frac{\ln(x)}{x} = 0$

Of course it is easy to find it by l'hopital's rule, but i want to find it using the power series expansion. But as power series of $\ln(x)$ is not nice, i need to use it's inverse, and that is $e^x$. Can you please explain me how this derivation works? Procedure and steps.

Many thanks

Answer 1

Let $x=e^t$. We are interested in the behaviour of $\frac{t}{e^t}$as $t\to\infty$.

Use the fact that for positive $t$ we have $e^t\gt 1+t+\frac{t^2}{2!}\gt \frac{t^2}{2}$.

Answer 2

Let's start with $\displaystyle\lim_{x\to\infty}\frac{\ln(x)}{x} = L$. Then we can take $\displaystyle\lim_{x\to\infty}e^\frac{\ln(x)}{x} = e^L$ which is the same as saying $\displaystyle\lim_{x\to\infty}(e^{\ln(x)})^\frac{1}{x} = e^L$. By properties of exponential and logarithm functions, we get $\displaystyle\lim_{x\to\infty}x^\frac{1}{x} = e^L$. Hopefully you know that $\displaystyle\lim_{x\to\infty}x^\frac{1}{x} = 1$. Thus $e^L = 1 \implies L=0$.