limit of $x \log y$ at $ (0,0)$

Question

What is the limit of the function $x \log y $ at $(0,0)$? I believe the limit doesn't exist. But wolframalpha.com says the limit is 0.

@MichaelAlbanese: The normal one, the limit of a 2-variable function. Not the iterated limit. — Chao, Dec 19 '13 at 09:27
Wolframalpha can report the limit doesn't exist for $$\lim_{(x,y)\rightarrow (0,0)} \frac{x}{\sqrt{x^2+y^2}}$$, but get it wrong for this one. — Chao, Dec 19 '13 at 09:31

André Nicolas · Answer 1 · 2013-12-19T06:00:58.050

2

You are right in being suspicious. Approach $(0,0)$ along the path $y=e^{-1/x}$. The limit along this path is $-1$. Modify to taste.

edited Dec 19 '13 at 06:00

answered Dec 19 '13 at 05:45

André Nicolas

Is this really a valid path? $y = e^{-1/x}$ doesn't contain the point $(0,0)$. – Adriano Dec 19 '13 at 05:50
@Adriano: You are right, it doesn't contain $(0,0)$. But as $x\to 0^+$, $e^{-1/x}\to 0$. That is what matters. – André Nicolas Dec 19 '13 at 05:53
Along the path y=x, the limit is 0. – dan Dec 19 '13 at 10:24

1 Answers1