let $f(x,y)= \frac{x^2|x|^{\beta}y}{x^4+y^2}$ when $x\neq0$ . we have to show that $f $ is continuous at $(0,0)$ for $\beta >0$

Question

let $$f(x,y)= \frac{x^2|x|^{\beta}y}{x^4+y^2}$$ when $x\neq0$ and $f(x,y)=0$ when $x=0$

we have to show that $f $ is continuous at $(0,0)$ for $\beta >0$

i am struggling to prove that. i have prove this for $\beta >1$. but how to do this for $0<\beta< 1$

Please provide some hint.

Answer 1

Using that $$x^4+y^2\geq 2x^2|y|$$ we get $$\frac{x^2x^{\beta}|y|}{x^4+y^2}\le \frac{x^2|x|^\beta|y|}{2|x|^2|y|}=\frac{x^\beta}{2}$$