Show that the limit does not exist $\lim_{(x, y) \to (0,0)}\frac{5x^2}{x^2 + y^2}$

Question

Show that the limit does not exist $\lim_{(x, y) \to (0,0)}\frac{5x^2}{x^2 + y^2}$

attempt:

let $y = 0$

$\lim_{x \to 0} \frac{5x^2}{x^2 + 0^2} = 5$

let $x = 0$

$\lim_{y \to 0} \frac{5(0)^2}{y^2} = 0$

$5 \neq 0$, therefore two different values, limit does not exist

right?

Answer 1

Yes, your proof is complete and you have explained your work clearly.