<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] $$ \sqrt{n+1}(\sqrt{n^3 + n} - \sqrt{n^3 + 1}) \cdot \frac{\sqrt{n^3 + n}+ \sqrt{n^3 + 1}}{\sqrt{n^3 + n}+ \sqrt{n^3 + 1}} = \frac{\sqrt{n+1}(n-1)}{\sqrt{n^3 + n}+ \sqrt{n^3 + 1}}$$ Now divide numerator and denominator by $\sqrt{n^3}.$
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