How can I solve the limit: $\lim_{n\to\infty} \frac{((-1)^n+5)^{1/n}}{3}$ How can I solve the limit ($n \in \mathbb{N}$): $$\lim_{n\to\infty} \frac{((-1)^n+5)^{1/n}}{3}$$Is it ok to say that it is greater or equal to $\frac{4^{1/n}}{3}$ and lower or equal to $\frac{6^{1/n}}{3}$ and thus the limit $=\frac{1}{3}$ ?

Split your sequence in two subsequences, corresponding to $n$ even and, respectively, $n$ odd. For even $n$, your quantity becomes $\dfrac {6^{\frac 1 n}} 3$ which, indeed, tends to $\frac 1 3$. For odd $n$, the quantity is $\dfrac {4^{\frac 1 n}} 3$ which, again, tends to $\frac 1 3$. Therefore, your quantity tends to $\frac 1 3$.
Your argument based on inequalities is equally valid.
