I need to solve the following problem: $\lim_{x\to 3}(x-3) \cot{\pi x}$. Can anyone give me a hint? I have no idea.
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You can transform to a limit to $0$ and express the cotangent as the well-known ratio. – Jan 19 '21 at 15:31
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Well the ratio is $\frac{cos{\pi x}}{sin{\pi x}}$. But how do I transform the limit? – Ilja Jan 19 '21 at 15:46
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1Shift the variable. – Jan 19 '21 at 15:58
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Can you show me how it is done, please? – Ilja Jan 19 '21 at 16:13
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$$\lim_{x\to 3}(x-3) \cot{\pi x}=\lim_{x\to 3} \frac{(x-3)\cos{\pi x}}{\sin{\pi x}}=\frac{0}{0}.$$.
Now, you can use L'Hopital rule, just one and then you will get your answer. I hope that helps
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$$\lim_{x\to 3}(x-3) \cot{\pi x}=-\lim_{x\to 0}x \cot{\pi x}=-\frac1\pi\lim_{x\to 0}\frac{\pi x}{\sin \pi x} \cos{\pi x}.$$
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Are you sure, that this is right because wolfram alpha says that it us now a different answer. – Ilja Jan 19 '21 at 17:24
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