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Trying to show that

$$\lim_{(x, y) \to (0, 0)}\dfrac{\sin(x) - \sin(y)}{x+y}$$

does not exist, but I'm having a lot of trouble. So far I've tried splitting the expression into two parts, but couldn't see that being conclusive. I've also tried L'Hospital's, but I'm not very experienced with derivatives yet.

Could someone help point me in the right direction to solve this? Thanks.

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  • $\begingroup$ Do not use $$ in titles. It clutters the front page. $\endgroup$ Commented Sep 29, 2014 at 22:04
  • $\begingroup$ Sorry, wasn't able to think of a compelling title without. Will take note for the future. $\endgroup$
    – user176049
    Commented Sep 29, 2014 at 22:41

3 Answers 3

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Hint: If you take the limit along the line $x=0$ what do you get? How about along the line $y=0$?

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  • $\begingroup$ Oh! Thanks, that makes it really obvious. When $x=0$, it's the negative of when $y=0$. It forms a sort of huge drop-off on its graph and the limit doesn't exist because it's not the same from all sides. $\endgroup$
    – user176049
    Commented Sep 29, 2014 at 22:44
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Hint: Check that the limit give different answers using different paths. Try making $x = y$ for one path, and $y=0$ for another.

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What happens near the line $y = -x$ What about the line $y = x$?

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