A Limit Problem

A Discord user wondered how to solve the following problem: determine $$\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{{10}^n+9^n+8^n}.$$

Solution.

Observe that $$\sqrt[n]{{10}^n+9^n+8^n}=10\sqrt[n]{1+{\left(\frac{9}{10}\right)}^n+{\left(\frac{8}{10}\right)}^n}$$ For convenience, let $a=9/10$ and $b=8/10$. Now we have $$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{{10}^n+9^n+8^n}& =10\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{1+a^n+b^n}\\ & =10\underset{n\to \mathrm{\infty }}{lim}\exp \log (1+a^n+b^n{)}^{1/n}\\ & =10\exp \underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\log (1+a^n+b^n)\end{array}$$ Since $a<1$ and $b<1$ we have $a^n\to 0$ and $b^n\to 0$ as $n\to \mathrm{\infty }$.
Now we utilize the small log approximation $\log (1+x)\approx x$ for sufficiently small $x$. (This can be justified by Taylor expanding $\log (1+x)$ around $0$.) $$\begin{array}{rl}10\exp \underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\log (1+a^n+b^n)& =10\exp \underset{n\to \mathrm{\infty }}{lim}\frac{a^n+b^n}{n}\\ & =10\exp (0)\\ & =10\end{array}$$ $◻$