Multivariable Limit - Squeeze Theorem

Question

I was trying to prove that this limit exists, but I cannot use Polar coordinates. Is there a special inequality that I can use along with the Squeeze Theorem to show that it does indeed equal $0$? Thanks.

$$\lim_{(x,y)\to(0,0)}\dfrac{(x^5y^2)}{(x^{10} + y^4)}.$$

This is just a substitution away from the standard $xy/(x^2+y^2)$ problem. — zhw., Sep 20 '15 at 17:55
Similar to https://math.stackexchange.com/questions/572125/show-discontinuity-of-fracxyx2y2 — Arnaud D., Dec 03 '19 at 16:24

score 2 · Answer 1 · answered Sep 20 '15 at 17:09

2

The limit doesn't exist. For instance, in the $x$-axis you always have $0$, but if on the curve $y=x^{\frac{5}{2}}$ you have always $\frac{1}{2}$.

answered Sep 20 '15 at 17:09

Sonner

3,108

Good enough. +1 – Mark Viola Sep 20 '15 at 18:41

Multivariable Limit - Squeeze Theorem

1 Answers1