Answer to Question #121883 in Calculus for Tuhin subhra Daa

Question

find fx(0,0) and fx(x,y) where (x,y) 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)?

Answer 1

$$f(x,y) = \begin{cases} \dfrac{xy^3 }{x^2 +y^2} &\text{if } (x, y)\not=(0, 0) \\ 0 &\text{if } (x,y)=(0,0) \end{cases}$$

If $(x, y)\not=(0, 0)$

$$f_{x}=\dfrac{\partial f}{\partial x}=\dfrac{\partial }{\partial x}( \dfrac{xy^3 }{x^2 +y^2} )=$$

$$=\dfrac{y^3(x^2+y^2)-2x^2y^3 }{(x^2 +y^2)^2}=\dfrac{y^3(y^2-x^2)}{(x^2 +y^2)^2}$$

$$f_x(0,0)=\lim\limits_{h\to 0}\dfrac{f(0+h,0)-f(0,0) }{h}=$$

$$=\lim\limits_{h\to 0}\dfrac{f(0+h,0)-f(0,0) }{h}=$$

$$=\lim\limits_{h\to 0}\dfrac{ \dfrac{h(0)^3 }{h^2 +(0)^2}-0 }{h}=0$$

$$f_x(x,y) = \begin{cases} \dfrac{y^3(x^2-y^2) }{(x^2 +y^2)^2} &\text{if } (x, y)\not=(0, 0) \\ 0 &\text{if } (x,y)=(0,0) \end{cases}$$

Let $x=rcos\theta, y=r\sin\theta$

$$\lim\limits_{(x, y)\to 0}f_x(x, y)=\lim\limits_{(x, y)\to 0}\dfrac{y^3(x^2-y^2) }{(x^2 +y^2)^2}=$$

$$=\lim\limits_{r\to 0}\dfrac{r^3\sin^3\theta r^2(\cos^2\theta-\sin^2\theta) }{r^4} =$$

$$=\lim\limits_{r\to 0}r\sin^3\theta(\cos^2\theta-\sin^2\theta)=0=f_x(0,0)$$

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