Answer to Question #105385 in Calculus for SHIVAM KUMAR

Question

Find fx (0,0) and fx (x,y),where (x,y)≠(0,0) for the following function
f (x,y)=xy^3/x^2+y^2, (x,y)≠(0,0)
0, (x,y)=(0,0)
is continuous at (0,0)? Justify your answer.

Answer 1

Let's find a limit of this function at (0,0)

$\lim\limits_{x,y\to 0}\frac{xy^3}{x^2+y^2}=[y=x^c;c\in R]=\lim\limits_{x,y\to 0}\frac{x^{3c+1}}{x^2(1+x^{2(c-1})}=0=f(0,0)$

Therefore the function is continuous at (0,0)