Question

Find f(0,0) and fr(x, y), where (x, y) = (0,0) for the following function
if x, y is equal to x y cube upon x square + y square if x, y does not is equal to 0, 0 and 0, x, y is equal 200

Answer 1

Given $f(x,y) = \begin{cases} \frac{xy^3}{x^2+y^2} : (x,y)\neq (0,0) \\ 0 \ \ \ \ \ \ \ : (x,y)=(0,0) \end{cases}$

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

And, $f_y(0,0) = \lim_{k \to 0} \frac{f(0,k)-f(0,0)}{k} = \lim_{h\to 0} \frac{0-0}{k} = 0$ .