Q:

query on limit of sequence

Suppose $a>0$, consider the sequence $a_n=n(\sqrt[n]{ea}-\sqrt[n]{a}), n\geq 1$ then $\lim_{n\to \infty} a_n=\text{ ?}$

It is given that limit tends to $1$ but I find it tends to $\infty$. Please explain how it tends to $1$. 

A:

Hint: $$a_n = \sqrt[n]{a}\cdot n(\sqrt[n]{e}-1).$$
It suffices to show that $\lim\limits_{n\to\infty}n(\sqrt[n]{e}-1) = 1$. If you write $$n(\sqrt[n]{e}-1) = \frac{e^{1/n}-e^0}{\frac 1n},$$ you should see the result.

