Let $a \in \mathbb{R}$ and $\lim b\_n = b$ (for a real sequence $b\_n$). I am trying to show that $\lim a^{b\_n}=a^b$ by using the definition of the convergence of a sequence, but I cannot see how to estimate $|a^{b\_n} - a^b|$ against $|b\_n - b|$ properly.