Answer
$\frac{\pi}{4}$.
Work Step by Step
Using the graph of the sequence, the value of $n$ approaches to $0.785$ as $n$ becomes larger.
We know $0.785\approx \frac{\pi}{4}$.
So, the sequence is convergent and we guess that the value of limit is $\frac{\pi}{4}$.
Now we give the prove:
$$\begin{aligned}
&\lim_{n\rightarrow \infty}\arctan\left(\frac{n^2}{n^2+1}\right)\\
&=\arctan\lim_{n\rightarrow \infty}\frac{n^2}{n^2+1}\\
&=\arctan 1\\
&=\frac{\pi}{4}.
\end{aligned}$$
