Question

Using polar coordinates find if the limit does or does not exist for

Lim(x,y)approaches(0,0) (2x/(x^2+X+y^2))

Answer 1

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

$$\dfrac{2x}{x^2+x+y^2}=\dfrac{2r\cos \theta}{r^2+r\cos \theta}=\dfrac{2\cos \theta}{r+\cos \theta}$$

The limit exists and

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

$$=\dfrac{2\cos \theta}{0+\cos \theta}=2$$