# Does limit of $|x|^{1/|y|}$ exist at $(0,0) ?$

Does limit of $$|x|^{1/|y|}$$ exist at $$(0,0) ?$$

This limit is zero along any path I've taken. But I am not able find out if the limit exists in general.

Any hint would be helpful.

• which paths have you taken? – peek-a-boo Jan 2 '20 at 11:06
• It’s like $0^ \infty$, why would it give any problem? I mean, what is bothering you? Maybe I am not seeing something – tommy1996q Jan 2 '20 at 11:08
• @peek-a-boo $0^\infty$ is not an indeterminate form. It's $0$. – bjorn93 Jan 2 '20 at 11:11
• @bjorn93 yes of course, that was indeed a very silly mistake (for some reason my brain did a different calculation) – peek-a-boo Jan 2 '20 at 11:14
• As a hint, note that by definition, $|x|^{1/|y|} = \exp\left( (\ln|x|)/|y| \right)$. What happens to $ln|x|$ as $x \to 0$? What happens to $|y|$ as $y \to 0$? What happens to the quotient (lol this is where I initially made a mistake)? What can you now conclude about the whole expression? – peek-a-boo Jan 2 '20 at 11:20

## 1 Answer

If $$|y|, |x|<1$$, then $$|x|^{\frac{1}{|y|}} <|x|$$. So for any path going to $$(0,0)$$, the expression is eventually bounded by $$|x|$$, which along the path goes to $$0$$, so the limit is indeed $$0$$.