Answer
Converges to $\frac{1}{3}$.
Work Step by Step
$\lim\limits_{n \to \infty}a_{n}=\lim\limits_{n \to \infty}\sqrt {\frac{n+1}{9n+1}}$
$=\lim\limits_{n \to \infty}\sqrt {\frac{1+1/n}{9+1/n}}$
$=\sqrt {\frac{1}{9}}$
$=\frac{1}{3}$
The given sequence converges to $\frac{1}{3}$.
