What is the limit of $3^{1/n}$ when n approaches infinity Graphically, I see that $\lim_{n->\infty}3^{1/n}$ approaches $1$. However, how to show $\lim_{n->\infty}3^{1/n} = 1$ step by step? 

Bernoulli's Inequality, which, for integer exponents, can be proven using a simple inductive argument, says
$$
\left(1+\frac2n\right)^n\ge3\ge1
$$
Taking roots, we get
$$
1\le3^{1/n}\le1+\frac2n
$$
Then, the Squeeze Theorem ensures that
$$
\lim_{n\to\infty}3^{1/n}=1
$$
