Q:

Evaluate $\lim_{x\to\infty}\left[x - \sqrt[n]{(x - a_1)(x - a_2)\ldots(x - a_n)}\right]$

Evaluate the following the limit:
$$\lim_{x\to\infty}\left[x - \sqrt[n]{(x - a_1)(x - a_2)\ldots(x - a_n)}\right]$$
I tried expressing the limit in the form $f(x)g(x)\left[\frac{1}{f(x)} - \frac{1}{g(x)}\right]$ but it did not help.

A:

\begin{align}
& \lim_{x\to\infty} x - \sqrt[n]{(x - a_1)(x - a_2)\ldots(x - a_n)} \\
= {} & \lim_{x\to\infty} x\left(1-\sqrt[n]{\left(1 - \frac{a_1}{x}\right)\left(1 - \frac{a_2}{x}\right)\ldots\left(1 - \frac{a_n}{x}\right)} \right) \\
= {} & \lim_{x\to\infty} x\left(1-1 +\frac{1}{n}\frac{\sum a_i}{x}\right)\\
= {} & \frac{1}{n}{\sum a_i}
\end{align}

