Show that $g(x,y)=\frac{x^2+y^2}{x+y}$ with $g(x,y)=0$ if $x+y=0$ is continuous at $(0,0)$. [closed]
Solution 1:
Hint: To find the limit at $(0,0)$ use polar coordinates $x=r\cos(\theta),$ $y=r\sin(\theta)$ and consider taking the limit as $r\to 0$.
Solution 2:
Use polar coordinates. Since $\sin \theta + \cos \theta = \sqrt{2} \sin(\theta + \frac{\pi}{4})$ then for $0 \leq \theta \leq \frac{\pi}{2}$ the denominator is clearly bounded away from zero on $M$.