If I only know $f'(x) = e^{x^2}$, how do I evaluate $\lim_{x \to \infty} \frac{f'(x)}{f(x)}$?

Question

Specifically, I need to show that it equals $\infty$. I remember writing the solution down somewhere, though I can't find it and can't remember it, so I'm mostly looking for an outline of how to solve it. Thanks in advance.

score 3 · Answer 1 · answered Feb 03 '14 at 06:36

3

Hint

I suppose you noticed that $\frac{f'(x)}{f(x)}$ is just the derivative of $\log (f(x))$.

This could help you goind faster to the solution.

answered Feb 03 '14 at 06:36

Claude Leibovici

260,315

score 2 · Accepted Answer · answered Feb 03 '14 at 03:23

2

Hint:

Use L'Hopital's rule to get something that you can immediately evaluate.

answered Feb 03 '14 at 03:23

Nathaniel Bubis

33,018

Thanks. Didn't recognize that was the right path since I wasn't sure how to show that $lim_{x \to \infty} f(x)$ was $\infty$, but I realized that it follows pretty easily from the MVT. – access Feb 03 '14 at 03:38
1

@access: as long as $f'(x)\ge1$, we get $\lim\limits_{x\to\infty}f(x)=\infty$. – robjohn Feb 03 '14 at 07:16

If I only know $f'(x) = e^{x^2}$, how do I evaluate $\lim_{x \to \infty} \frac{f'(x)}{f(x)}$?

2 Answers2