Sequence involving $\sqrt[n]{n}$ and $\text{log}$ I know that the sequence $\sqrt[n]{n}$ converges to 1 and that $\text{log}(\sqrt[n]{n})$ thus converges to 0 as $n\to\infty$ since the logarithmic function is continuous. But how can I calculate the limits as $n\to\infty$ of the following sequences?
(a) $\frac{\sqrt[n]{n^2}-1}{\text{log}(\sqrt[n]{n})}$
(b) $\frac{\sqrt[n]{1/n}-1}{\text{log}(\sqrt[n]{n})}$
Any help would be much appreciated!

(a) Write this quotient as
$$\frac{e^{(1/n)\ln n^2}-1}{(1/n)\ln n} = \frac{e^{2(1/n)\ln n}-1}{(1/n)\ln n}.$$
As $n\to \infty,$ $(1/n)\ln n\to 0.$ So the limit of the above has the form
$$\lim_{u\to 0}\frac{e^{2u}-1}{u}.$$
That should look familiar.
