Why is $\lim_{(x_1,x_2) \to (0,0)}\frac{x_1x_2}{x_1^2 + x_2^2} ≠ 0$

Question

I have tried solving the equation but I keep getting zero. I also can't find a proof online, I think because it is a rather specific question.

See https://math.stackexchange.com/questions/572125/show-discontinuity-of-fracxyx2y2? — Robert Z, May 08 '19 at 09:11

score 1 · Answer 1 · answered May 08 '19 at 09:08

1

I have tried solving the equation but I keep getting zero.

Hint

What happens if $x_1=x_2$, so when approaching the origin along that line?

answered May 08 '19 at 09:08

StackTD

score 1 · Answer 2 · answered May 08 '19 at 09:08

1

Just take $x_2=x_1$ and the limit will be $\frac12$. But if you take $x_2=0$, the limit is $0$. Therefore, the limit doesn't exist.

answered May 08 '19 at 09:08

score 0 · Answer 3 · answered May 08 '19 at 09:08

0

Put $x_1=x_2=\frac 1 n$ to see that the limit is not $0$.

answered May 08 '19 at 09:08

Kavi Rama Murthy

score 0 · Answer 4 · answered May 08 '19 at 09:18

0

$\dfrac{x_2/x_1}{1+x_2^2/x_1^2}$;

Let $m:= x_2/x_1$, then?

answered May 08 '19 at 09:18

Peter Szilas

4 Answers4