Question

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

Answer 1

Given

$f(x,y) = \frac{4x^2y }{4x^4+y^2}$

To evaluate

$\lim_{(x,y)\to(0,0)}f(x,y) = \lim_{(x,y)\to(0,0)}\frac{4x^2y }{4x^4+y^2}$

Choose path passes through (0,0), let

$y=mx$

then the limit along y=mx given by

$\begin{aligned} \lim_{(x,y)\to(0,0)}\frac{4x^2y }{4x^4+y^2}&= \lim_{x\to0}\frac{4mx^3 }{4x^4+m^2x^2}\\ &= \lim_{x\to0}\frac{4mx }{4x^2+m^2 }\\ &= \lim_{x\to0}\frac{0 }{0+m^2 }=0 \end{aligned}$

Since the limit does not depend on m , then limit exist and equal to 0

$\lim_{(x,y)\to(0,0)}f(x,y) =f(0,0)=0$

Then f(x,y ) is continuous at (0,0)