Prove continuity by limit definition.

Question

I had a Calc$2$ exam today and I was given this function:

$$ f(x,y)=\frac {xe^{-1/x^2}\sin(1/y)}{x^2+y^2} \text{ if xy $\neq0$} $$ And $f=0$ if $xy=0$.

I was asked to prove $f$ is continuous at $0$ and I couldn't do it, I thought maybe getting some nicer bounding functions would help out but I couldn't think of any that did.

Answer 1

Hint: $\vert \frac{x e^\frac{-1}{x^2}sin\frac{1}{y}}{x^2+y^2}-0\vert\leq \vert e^\frac{-1}{x^2}\vert$ and $\lim_{x\to o}e^\frac{-1}{x^2}=\lim_{x\to o}\frac{1}{e^{x^2}}=\frac{1}{e^\frac{1}{0^2}}=\frac{1}{e^{+\infty}}=0$

Note that $\vert \frac{x sin\frac{1}{y}}{x^2+y^2}\vert\leq 1$

Answer 2

Hint(s):

• $|\sin(u)|\le 1$ for all $u\in \mathbb{R}$.

• $\frac{|x|}{|x^2+y^2|}\le \frac{1}{|x|}$ if $x\neq 0$.