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## How to find this limits?

Can someone help me with this limit? $$\lim_{(x,y)\to (0,0)} \frac{\sin(x-y)}{x+y}$$ Over the domain $x>0$, $y>0$. I think the result is (zero) but I can’t conclude any estimation due to $|x+y|$ at denominator.

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## Limit of a sum of squared sines

Following up on this question Nice Limit $\lim_{n\to\infty}\sum_{k=1}^{n} \sin^2\left(\frac{\pi}{n+k}\right)$ . Fix a real $a$ and consider the sum: $S(n)=\sum_{k=1}^n sin^2\left(\frac{n^a\pi}{n+k}\right)$ In the linked question it is shown that for $a=0$ the sum converges to zero for large $n$. It is also easy to show that for $a=1$ the sum diverges, recognizing a Riemann sum. The […]

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