What is the value of this limit? $$ \lim_{x \to \infty}x^{\frac{1}{x}} $$
I have never encountered such a limit before, so any help or advice would be much appreciated.
Asked
Active
Viewed 356 times
5
-
What's the definition of $x\mapsto x^{1/x}$? – Git Gud Jun 29 '14 at 13:10
-
7What is it, sir? – user34304 Jun 29 '14 at 13:12
-
1It is $e^{\frac {\log(x)}x}$, for all $x>0$. It's undefined for $x\leq 0$. – Git Gud Jun 29 '14 at 13:13
-
4Funny. Why did two people up vote the OP's question in the comment? – Git Gud Jun 29 '14 at 13:20
-
Two people didn't know it either, I guess. – Alex Jun 29 '14 at 13:22
-
@Alex The thing is the up votes actually came after my reply to that comment. – Git Gud Jun 29 '14 at 13:23
-
1Then I guess they're glad the OP asked? – G Tony Jacobs Jun 29 '14 at 13:25
-
See also http://math.stackexchange.com/questions/1092451/need-help-finding-lim-x-rightarrow-infty-sqrtxx and http://math.stackexchange.com/questions/2105965/prove-lim-x-rightarrow-infty-sqrtxx-1-no-lh%C3%B4pital – Martin Sleziak Jan 22 '17 at 14:33
-
@GitGud because it seems like he is mocking you – Black Jack 21 Jan 22 '17 at 14:35
3 Answers
10
Here's a start:
$\lim x^{1/x} = \lim \exp(\log(x^{1/x)})) = \lim \exp\left[\frac1x\log x\right] = \exp\left[\lim\frac1x\log x\right]$.
The limit in that last expression is a $0\cdot\infty$ form. Do you know how to handle those with L'Hôpital's Rule?
G Tony Jacobs
- 32,044
4
Hint
Use the L'Hôpital's rule to find
$$\lim_{x\to\infty}\frac{\ln x}{x}$$
2
An approach similar to G Tony Jacobs: use the continuity of logarithm (i.e. $\log \lim f(x) = \lim \log f(x)$) to log the expression to get $$ L f(x) = \frac{\log x}{x} $$ then show it converges to $0$ by L'Hospital's rule, then exponentiate back to get 1.
Alex
- 19,395