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I already know that the value of the limit $$\lim_{n \to 0} n\cot(n)$$ is equal to $1$, but I'm not quite sure how to evaluate it in an airtight way. I tried L'Hopital, but it just becomes circular.

Does anybody have any hints?

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$$\lim\limits_{n\rightarrow 0 } n \frac{\cos n }{\sin n}$$

Use $\lim\limits_{n\rightarrow 0} \frac{\sin n}{n} = 1$.

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