Rewriting the limit $e^{f(x)}$

Question

I've seen in multiple places that a limit

$\lim_{x\to\infty} e^{f(x)}$

can be rewritten as

$e^{\lim_{x\to\infty} f(x)}$.

However, I searched Google (and this Stack Exchange) for limit properties, but none of them seem to state this as a rule or imply it, either. Is there a way to prove this, and is there a rule which states this?

score 4 · Accepted Answer · edited Mar 25 '16 at 08:13

4

If $\lim_{x\to \infty }f(x)$ exists, the continuity of $x\longmapsto e^x$ gives you the result directly. If $\lim_{x\to \infty }f(x)=+\infty $, it's a fact that $x\longmapsto e^x$ is strictly increasing and that $\lim_{x\to \infty }e^{x}=+\infty $ will give you the result. And if $\lim_{x\to \infty }f(x)$ doesn't exist, then it makes no sense to write $\lim_{x\to\infty }f(x)$.

edited Mar 25 '16 at 08:13

Skeleton Bow

2,101

answered Mar 25 '16 at 08:03

Surb

55,662

Thanks! What about for any number, not just for infinity? – Skeleton Bow Mar 25 '16 at 08:04
what do you mean ? @SkeletonBow – Surb Mar 25 '16 at 08:06
I really don't know – Skeleton Bow Mar 25 '16 at 08:10
1

????? @SkeletonBow – Surb Mar 25 '16 at 08:11
1

@SkeletonBow Nothing strange will happen then. If $\lim_{x\rightarrow\infty} f(x)=A$, then we get that $\lim_{x\rightarrow\infty} e^{f(x)}=e^{\lim_{x\rightarrow\infty} f(x)}=e^A$. It's a direct consequence of the continuity of the exponential function. – Scounged Mar 25 '16 at 08:23
@Scounged superb. Thank you very much! – Skeleton Bow Mar 25 '16 at 08:24

Rewriting the limit $e^{f(x)}$

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