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The limit in question is

$$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$$

I set $z=0$ and converted this to polar coordinates:

$$\lim_{r\to0}\frac{r^2\cos \theta \sin \theta}{r^2\cos^2 \theta +r^2\sin^2 \theta}$$

This can be reduced to $$\lim_{r\to0}\cos \theta \sin \theta$$

if I did everything properly. Does this mean that the limit on the origin doesn't exist? Plugging different values of $\theta$ in the limit returns different values, but I'm not entirely sure if this is correct. I know there are easier ways to do this, and several of them are already on MSE, but I have a feeling a trick like this (if it is correct) might be useful for some other more complicated limit.

user401936
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  • You are correct, the limit does not exist. – Rob Mar 12 '17 at 20:14
  • That's the topic I was talking about in the original post, I was just curious to see if my more complicated way of doing the exercise was correct. – user401936 Mar 12 '17 at 21:39

1 Answers1

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Yes because it depends on theta it does not exist.

Consider approaching the origin with $\vec r(t)=\langle t,t,0 \rangle$.

We get,

$$\lim_{t \to 0} \frac{t^2}{2t^2}=\lim_{t \to 0} \frac{1}{2}=\frac{1}{2}$$

Then consider approaching with $\vec r(t)=\langle 0,0,t \rangle$.

We get,

$$\lim_{t \to 0} \frac{0}{t^4}=0$$

Therefore the limit can't exist.

Ahmed S. Attaalla
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