Question

 Discuss the continuity of f(x, y) = (x^4)(y^4)/(x2+y 2)^3 if (x, y) not equal to (0,0), 0 if (x, y) = (0,0) at(0, 0) 

Answer 1

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

$$\dfrac{x^4y^4}{(x^2+y^2)^3}=\dfrac{(r^4\cos^4\theta) (r^4\sin^4\theta)}{(r^2\cos^2\theta+r^2\cos^2\theta)^3}=$$

$$=r^2\cos^4\theta\sin^4\theta$$

$$\lim\limits_{(x,y)\to(0,0)}\dfrac{x^4y^4}{(x^2+y^2)^3}=\lim\limits_{r\to0}(r^2\cos^4\theta\sin^4\theta)=0$$

We have that

$$\lim\limits_{(x,y)\to(0,0)}f(x,y)=0=f(0,0)$$

Therefore the function

$$f(x, y)= \begin{cases} \dfrac{x^4y^4}{(x^2+y^2)^3} &\text{if } (x,y)\not=(0,0) \\ 0 &\text{if } (x,y)=(0,0) \end{cases}$$

is continuous at $(0,0).$