## Task 33/78

Determine the limit $\lim _{n \rightarrow \infty} \sqrt[n]{a^{n}+b^{n}}$, where $a$ and $b$ are arbitrary positive real numbers with $a \geq b$.
From $0<b \leq a$ it follows that $0<b^{n} \leq a^{n}$. Therefore, we have

$$
\begin{gathered}
a^{n}<a^{n}+b^{n} \leq 2 a^{n} \quad ; \quad a<\sqrt[n]{a^{n}+b^{n}} \leq a \sqrt[n]{2} \\
a=\lim _{n \rightarrow \infty} a \leq \lim _{n \rightarrow \infty} \sqrt[n]{a^{n}+b^{n}} \leq \lim _{n \rightarrow \infty} a \sqrt[n]{2}=a
\end{gathered}
$$

because $\lim _{n \rightarrow \infty} \sqrt[n]{a}=1$. This immediately implies

$$
\lim _{n \rightarrow \infty} \sqrt[n]{a^{n}+b^{n}}=a
$$