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.

  • $\begingroup$ which paths have you taken? $\endgroup$ – peek-a-boo Jan 2 '20 at 11:06
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    $\begingroup$ It’s like $0^ \infty$, why would it give any problem? I mean, what is bothering you? Maybe I am not seeing something $\endgroup$ – tommy1996q Jan 2 '20 at 11:08
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    $\begingroup$ @peek-a-boo $0^\infty$ is not an indeterminate form. It's $0$. $\endgroup$ – bjorn93 Jan 2 '20 at 11:11
  • $\begingroup$ @bjorn93 yes of course, that was indeed a very silly mistake (for some reason my brain did a different calculation) $\endgroup$ – peek-a-boo Jan 2 '20 at 11:14
  • $\begingroup$ 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? $\endgroup$ – peek-a-boo Jan 2 '20 at 11:20

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 $0$.


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