Prove or disprove each of the follow function has limits $x \to a$ by the definition $\lim_{(x, y) \to (0, 0)} \frac{xy^2}{x^2 + y^2}$

Question

Prove or disprove each of the follow function has limits $x \to a$ by the definition

$\lim_{(x, y) \to (0, 0)} \frac{xy^2}{x^2 + y^2}$

Given $\epsilon > 0$. Choose $\delta = \epsilon$, then $||(x, y) - (0, 0)|| = \sqrt{x^2+y^2}$, and so $\|(x, y) - (0,0)\| < \delta$ implies that

$$\left| \frac{xy^2}{x^2-y^2} - 0\right| = \frac{|xy^2|}{x^2+y^2} \leq \frac{\sqrt{x^2+y^2} \cdot (x^2+y^2)}{x^2 + y^2} = \sqrt{x^2+y^2} < \delta = \epsilon$$

Therefore the limit does exist.

Is this right?

Answer 1

You can also use this. $$\left|{xy^2\over x^2 + y^2 }\right| = |x| {y^2\over x^2 + y^2 } \le |x|. $$