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