What is wrong with this proof of $\lim_{(x,y)\to(0,0)}\frac{xy}{x^2+y^2}=0$

Question

I know that $\lim_{(x,y)\to(0,0)}\frac{xy}{x^2+y^2}$ cannot exist because taking $x=y$ and $x=-y$ gives me two different values for the limit.

However, I can also write the following:

$$|\frac{xy}{x^2+y^2}| \leq |xy|$$

This seems to suggest that the limit does exist because $|xy|\to 0$ as $(x,y)\to(0,0)$. Can someone explain what is happening here?

Answer 1

Your inequality is wrong because near $(0,0)$, $x^2 + y^2$ is less than $1$.

Answer 2

We have that

$$\frac{|xy|}{x^2+y^2}\ge |xy|$$

indeed recall that $x^2+y^2 \to 0$ and then it is less than $1$.

Answer 3

Because for $x$ and $y$ small enough, we have:

$$|x^2+y^2|<1\Rightarrow\left|\frac{1}{x^2+y^2}\right|>1\Rightarrow\left|\frac{xy}{x^2+y^2}\right|>|xy|$$