Answer to Question #288850 in Calculus for Pankaj

Question

The function f(x, y) ={x²y/x⁴ +y² , (x, y) ≠0 and 0 , (x, y) =0 }is not continuous at (0,0).

Say true or false.

Answer 1

Evaluate $\lim\limits_{(x,y)\to(0,0)}f(x, y)$ along path $y=mx$

$$\lim\limits_{(x,mx)\to(0,0)}\dfrac{x^2(mx)}{x^4+(mx)^2}=\lim\limits_{(x,mx)\to(0,0)}\dfrac{mx}{x^2+m^2}=0$$

Evaluate $\lim\limits_{(x,y)\to(0,0)}f(x, y)$ along path $y=kx^2$

$$\lim\limits_{(x,kx^2)\to(0,0)}\dfrac{x^2(kx^2)}{x^4+(kx^2)^2}=\lim\limits_{(x,kx^2)\to(0,0)}\dfrac{k}{1+k^2}=\dfrac{k}{1+k^2}\not=0$$

Therefore $\lim\limits_{(x,y)\to(0,0)}f(x, y)$ does not exist.

Therefore the function $f(x, y)$ is not continuous at $(0,0).$

The statement is False.