A Limit Problem

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

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 $$ \begin{equation} \begin{split} \lim_{n\rightarrow\infty}\sqrt[n]{10^n+9^n+8^n} &= 10\lim_{n\rightarrow\infty}\sqrt[n]{1 + a^n + b^n} \\ &= 10\lim_{n\rightarrow\infty}\exp\log(1+a^n+b^n)^{1/n} \\ &= 10\exp\lim_{n\rightarrow\infty}\frac{1}{n}\log(1+a^n+b^n) \end{split} \end{equation} $$ Since \(a<1\) and \(b<1\) we have \(a^n \rightarrow 0\) and \(b^n\rightarrow 0\) as \(n\rightarrow\infty\).
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{equation} \begin{split} 10\exp\lim_{n\rightarrow\infty}\frac{1}{n}\log(1+a^n+b^n) &= 10\exp\lim_{n\rightarrow\infty}\frac{a^n+b^n}{n} \\ &= 10\exp(0) \\ &= 10 \end{split} \end{equation} \(\square\)