<s>[INST] $\lim_{n \rightarrow \infty} \sqrt{n+1}(\sqrt{n^3 + n} - \sqrt{n^3 + 1})$ I wish to compute the following limit
$$\lim_{n \rightarrow \infty} \sqrt{n+1}(\sqrt{n^3 + n} - \sqrt{n^3 + 1})$$
Wolfram Alpha gives the answer as $1/2$. I've tried L'Hopital's rule, but I can't seem to get it to work.
 [/INST] Try
$$\lim_{n \rightarrow \infty} \sqrt{n+1}(\sqrt{n^3 + n} - \sqrt{n^3 + 1})= \lim_{n \rightarrow \infty} \sqrt{n+1}\frac{n-1}{\sqrt{n^3 + n} + \sqrt{n^3 + 1}}=\lim_{n \rightarrow \infty} \sqrt{1+1/n}\frac{1-1/n}{\sqrt{1 + 1/n^2} + \sqrt{1 + 1/n^3}}=1/2$$
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