<s>[INST] Question about $\lim_{n\rightarrow\infty} (\sqrt{n} - \sqrt{n - n^c})$ and $\lim_{n\rightarrow\infty} (\sqrt[3]{n + n^c} - \sqrt[3]{n})$ I have following exercises:

1) Find $\lim_{n\rightarrow\infty} (\sqrt{n} - \sqrt{n - n^c})$, $0\leq c\leq 1$
hint: make a case distinction for $c < \frac{1}{2}, c = \frac{1}{2}, c > \frac{1}{2}$
2) Find $\lim_{n\rightarrow\infty} (\sqrt[3]{n + n^c} - \sqrt[3]{n})$, $0\leq c\leq 1$
hint: make a case distinction for $c < \frac{2}{3}, c = \frac{2}{3}, c > \frac{2}{3}$


My question: without the hints, how could I derive the values $\frac{1}{2}$ and $\frac{2}{3}$ on my own?
 [/INST] $$\sqrt{n}-\sqrt{n-n^c}=\frac{n^c}{\sqrt{n}+\sqrt{n-n^c}}=\frac1{\sqrt{n^{\color{blue}{1-2c}}}+\sqrt{n^{1-2c}-n^{-c}}}$$
\begin{align}\sqrt[3]{n+n^c}-\sqrt[3]{n}&=\frac{n^c}{\sqrt[3]{(n+n^c)^2}+\sqrt[3]{n^2}+\sqrt[3]{n(n+n^c)}}\\&=\frac{1}{\sqrt[3]{(n+n^c)^2n^{-3c}}+\sqrt[3]{n^{\color{blue}{2-3c}}}+\sqrt[3]{n^{1-3c}(n+n^c)}} \end{align}
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