Answer
\[
\begin{array}{l}
\frac{\pi}{2}
\end{array}
\]
Work Step by Step
We have to find the limit:
\[
\begin{array}{l}
\lim _{(x, y) \rightarrow(0,1)} \tan ^{-1} \frac{1+x^{2}}{x^{2}+(y-1)^{2}} \\
=\lim _{(x, y) \rightarrow(0,1)} \tan ^{-1} \frac{x^{2}+1}{x^{2}+(y-1)^{2}} \\
=\tan ^{-1} \frac{0^{2}+1}{0^{2}+(1-1)^{2}} \\
=\tan ^{-1}\left(\frac{1}{0}\right) \\
=\tan ^{-1}(\infty) \\
=\frac{\pi}{2} \\
\frac{\pi}{2}
\end{array}
\]
