Answer to Question #197402 in Calculus for Sadeen Khan

Question

Find fx(0,0) and fx(x,y), where (x,y) not = (0,0) for the function f:R^2 is to R defined by

F(x,y)={xy^3/x^2+y^2 if (x,y) is not= (0,0) and (x,y)=(0,0).

Is Fx continuous at (0,0)? Justify your answer.

Answer 1

$$f_x(0, 0)=\lim\limits_{h\to0}\dfrac{f(0+h, 0)-f(0,0)}{h}$$

$$=\lim\limits_{h\to0}\dfrac{\dfrac{0\cdot(h)^3}{(0)^2+h^2}-0}{h}=0$$

$$f_y(0, 0)=\lim\limits_{h\to0}\dfrac{f(0, 0+h)-f(0,0)}{h}$$

$$=\lim\limits_{h\to0}\dfrac{\dfrac{h\cdot(0)^3}{h^2+(0)^2}-0}{h}=0$$

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

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

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

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

$$=\lim\limits_{r\to0}\dfrac{r^3 \sin^3\theta(r^2 \sin^2\theta-r^2 \cos^2\theta)}{r^4}$$

$$=\lim\limits_{r\to0}\dfrac{r\cos\theta (r\sin\theta)^3}{(r\cos\theta)^2+(r\sin\theta)^2}$$

$$=\lim\limits_{r\to0}r^2\cos\theta \sin^3\theta=0=f(0, 0)$$

$$=\lim\limits_{r\to0}r\sin^3\theta(\sin^2\theta- \cos^2\theta)=0=f_x(0, 0)$$

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