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.

Write $p(x) = (x-a_{1})(x-a_{2})\cdots (x-a_{n})$. Now
$$
\lim_{x\to\infty}[x - \sqrt[n]{(x - a_1)(x - a_2)\ldots(x - a_n)}] = \lim_{x \to \infty}[x-\sqrt[n]{p(x)}] \\
= \lim_{x \to \infty}\left((x-\sqrt[n]{p(x)})\frac{x^{n-1}+x^{n-2}\sqrt[n]{p(x)}+\ldots+x(\sqrt[n]{p(x)})^{n-2}+(\sqrt[n]{p(x)})^{n-1}}{x^{n-1}+x^{n-2}\sqrt[n]{p(x)}+\ldots+x(\sqrt[n]{p(x)})^{n-2}+(\sqrt[n]{p(x)})^{n-1}}\right) \\
= \lim_{x \to \infty}\left(\frac{x^{n}-p(x)}{x^{n-1}+x^{n-2}\sqrt[n]{p(x)}+\ldots+x(\sqrt[n]{p(x)})^{n-2}+(\sqrt[n]{p(x)})^{n-1}}\right) \\
= \lim_{x \to \infty}\left(\frac{(a_{1}+a_{2}+\ldots+a_{n})x^{n-1}+q(x)}{x^{n-1}\left(1+\frac{\sqrt[n]{p(x)}}{x}+\ldots+\frac{(\sqrt[n]{p(x)})^{n-2}}{x^{n-2}}+\frac{(\sqrt[n]{p(x)})^{n-1}}{x^{n-1}}\right)}\right) \\
= \lim_{x \to \infty}\left(\frac{a_{1}+a_{2}+\ldots+a_{n}+\frac{q(x)}{x^{n-1}}}{1+\frac{\sqrt[n]{p(x)}}{x}+\ldots+\left(\frac{\sqrt[n]{p(x)}}{x}\right)^{n-2}+\left(\frac{\sqrt[n]{p(x)}}{x}\right)^{n-1}}\right)
$$
where $q(x)$ is a polynomial of degree at most $n-2$.
Now $\frac{\sqrt[n]{p(x)}}{x} \to 1$, and $\frac{q(x)}{x^{n-1}} \to 0$ as $x \to \infty$ so the whole expression tends to
$$
\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}.
$$
