What is limit of $e^{x}/\ln(x)$ as $x \rightarrow +\infty$

Question

What is limit of $e^{x}/\ln(x)$ as $x \rightarrow +\infty$

My professor says that the answer is $+\infty$ but when I try it I get $\infty/\infty$ which is an indeterminate form.

How to get sense of that?

Answer 1

Basically, it's because $\color{darkblue}{e^x}$ increases way faster than $\color{darkmagenta}{\ln x}$.

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I hope this helps.
Best wishes, $\mathcal H$akim.

Answer 2

By L'Hôpital's rule:

$$\lim_{x\rightarrow \infty} \frac{e^x}{\ln x} = \lim_{x\rightarrow\infty} xe^x = \infty$$

Answer 3

$e^x$ is asymptotically larger than any polynomial and the logarithm is smaller.