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What is the result of $\lim_{(x,y)\to (0,0)}\frac{x+y}{\sqrt{x^2+y^2}}$ . I tried to do couple of algebraic manipulations, but I didn't reach to any conclusion.

Thanks a lot.

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Have you tried writing this in polar coordinates so that you take the limits as $r$ goes to $0$? – Alex Becker Jan 23 '12 at 7:23
Try to set $ y = x $ and $ y = -x \ $ – WLOG Jan 23 '12 at 7:26
What stef said, and/or $y=0$. – Did Jan 23 '12 at 7:31
Generally speaking, the first two comments are fairly good approaches to problems like this. – AD. Jan 23 '12 at 7:33
up vote 5 down vote accepted

If $$x=r\cos\theta$$ $$y=r\sin\theta$$ then we have $$\text{lim}_{r\rightarrow 0}\frac{r\cos\theta+r\sin\theta}{r} =\cos\theta+\sin\theta=\sqrt{2}\sin(\theta+\frac{\pi}{4})$$ which depends on the angle of approach to the origin.

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