Answer to Question #107182 in Calculus for Nikesh gautam pandit ji

Question

Check whether the function
f(x,y)= { 4x^2y/4x^4+y^2,. (x,y)not=to(0,0)
0 (x,y) = to (0,0)}

is continuous at ( 0,0).

Answer 1

$\lim\limits_{(x;y)\to (0;0)}\frac{4x^2y}{4x^4+y^2}=[y=x^2]=\\=\lim\limits_{x\to 0}\frac{4x^2x^2}{4x^4+x^4}=\frac{4}{5}\ne0=f(0,0)$

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