<s>[INST] limit, quotient of roots How can I find
$$
\lim_{n \to \infty} \frac{\sqrt[3]{n + 2} - \sqrt[3]{n + 1}}{\sqrt{n + 2} - \sqrt{n + 1}} \sqrt[6]{n - 3}?
$$
If I multiply by $\sqrt{n + 2} + \sqrt{n + 1}$ I could get no divisor, but I cannot get any result on this way. On the other hand I could also transform this expression:
$$
\frac{(n + 2)^{1 / 3} \left(1 - {\left(\frac{n + 1}{n + 2}\right)}^{1 / 3}\right)}{(n + 2)^{1 / 2} \left(1 - {\left(\frac{n + 1}{n + 2}\right)}^{1 / 2}\right)} \sqrt[6]{n - 3} = {\left(\frac{n - 3}{n + 2}\right)}^{1 / 6} \frac{1 - {\left(\frac{n + 1}{n + 2}\right)}^{1 / 3}}{1 - {\left(\frac{n + 1}{n + 2}\right)}^{1 / 2}}.
$$
However I also cannot see how continue to see that the limit is $\frac 2 3$. Any help would be appreciated.
 [/INST] Put $ m=n+1$.
The limit becomes after factorisation and simplification
$$\lim_{m\to\infty}\frac{(1+\frac 1m)^{\frac 13}-1}{(1+\frac 1m)^{\frac 12}-1}(1-\frac 4m)^{\frac 16}$$
Use the result
$$\lim_{N\to \infty}N\Bigl((1+\frac 1N)^\alpha-1\Bigr)=\alpha$$
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