Answer to Question #115572 in Calculus for Neha

Question

Find Fx (0,0) and fx (x,y), where (x,y) is not equal to (0,0) for the following function
F(x,y)={xy^3/x^2+y^2, (x,y) not equal to (0,0)
0, (x,y)= (0,0)} is fx continuous at (0,0) justify your answer.

Answer 1

Given function is $f(x,y)=\frac{xy^3}{x^2+y^2}$ when $(x,y)\neq (0,0).$

For this case, denominator is not zero and also enumerator and denominator are continuous, so given function is differential for this case.

So, $f_x(x,y) = \frac{(x^2+y^2)(y^3)-(xy^3)(2x)}{(x^2+y^2)^2} = \frac{y^3(y^2-x^2)}{(x^2+y^2)^2}.$

Given $f(x,y)=0$ for $(x,y)=(0,0)$

Now $f_x(0,0) = \lim_{h\to0} \frac{f(0+h,0)-f(0,0)}{h} = \lim_{h\to0} \frac{0}{h} = 0$ .

Also, $\lim_{(x,y)\to(0,0)} f_x(x,y) = 0$ since numerator power is more than enumerator.

So, $f_x(x,y)$ is continuous at (0,0).