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Is there any good way to find a multivariable limit other than switching to polar coordinates?

For example, students each year are inundated with problems like $$\lim_{(x,y)\to (0,0)}{\frac{x^2y-xy^2}{\sqrt{x^2+y^2}}}$$ and, putting aside using the definition of the limit directly, the only good solution that seems to exist is to write $$\lim_{r\to 0}{\frac{r^3(\cos^2\theta\sin\theta-\cos\theta\sin^2\theta)}{r}}=\lim_{r\to 0}{\;r^2\cos\theta\sin\theta (\cos\theta -\sin\theta)}=0$$

Also, is there any instance in which this method fails? Don't feel obligated to provide a full-blown proof, I'm just wondering if there is a proof because I don't recall ever seeing one in a textbook.

  • The "polar coordinates" technique is used in such exercises because it makes it much easier to see whether there is "directional dependence" in approaching the origin (frequently the limit point in these problems) than to work in rectangular coordinates. One can use the substitution $ \ y = mx \ $ and consider whether the limit depends on the slope of the line along which the origin is approached, but there are functions for which using a line, or even a "power-function" $ \ y = cx^n \ $ , isn't conclusive. The "radial factor" in the polar technique often settles matters. – colormegone Apr 19 '14 at 04:02

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The polar coordinates trick usually works nicely if the denominator is a relative of $x^2+y^2$. This happens fairly often in exercises.

There certainly are other methods. For example, let $w=\max(|x|,|y|)$. Then the denominator is $\ge w$. The numerator has absolute value $\le 2w^3$. Thus the ratio is $\le 2w^2$, which approaches $0$ as $(x,y)\to (0,0)$.

André Nicolas
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Approaching the limit along a straight line does not always work! See this counterexample

Community
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Mark Perlman
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    Letting $r\to 0$ in polar does not amount to only considering straight lines. The angular variable $\theta$ can do absolutely anything, so all possible curves are taken into account. –  Aug 07 '14 at 22:42
  • You're absolutely right, I apologize; I was anticipating an error I tend to see a lot. – Mark Perlman Aug 09 '14 at 03:06