[Math] Why doesn’t $ \lim_{(x,y)\rightarrow(0,0)} xy\sin(\frac{1}{xy}) $ exist

limitsmultivariable-calculuswolfram alpha

I've tried to get the following limit:

$$ \lim_{(x,y)\rightarrow(0,0)} xy\sin\left(\frac{1}{xy}\right) $$

wolfram claims it doesn't exist.

  1. How to show that?
  2. Why can't I take $z=xy$ and receive a known limit of one variable?

$$ \lim_{z\rightarrow0} z \sin\left(\frac{1}{z}\right) $$

Best Answer

1) The limit clearly exists since $\sin$ is bounded and $xy\rightarrow 0$.

2) In this problem you can. In a problem such as $\lim_{(x,y)\rightarrow (0,0)}\frac{x}{y}$ you run into trouble because you seem to suggest that it's ok to write it as $\lim_{z\rightarrow 0}\frac{z}{z}=1$ and that's wrong. To see why it's wrong, $x/y$ can be arbitrarily large if you keep $x$ fixed and make $y$ smaller.

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