Does the limit exits if $ \lim_{n \to \infty} \frac{b_n}{n}$ if $b_n$ is monotone increasing?

Question

Suppose that $b_n$ is monotone increasing in $n$.

Question: Does the following limit exits (in the extended sense)? \begin{align} \lim_{n \to \infty} \frac{b_n}{n} \end{align}

If not, I would like to see an example that shows that $\limsup$ and $\liminf$ are different?