If a point \( P(x, y) \) lies on the curve \( y=f(x) \) such that \...

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If a point \( P(x, y) \) lies on the curve \( y=f(x) \) such that
\[
\lim _{(x, y) \rightarrow(1,2)}\left[\frac{\tan ^{-1} x+\tan ^{-1} \frac{1}{y}-\tan ^{-1} 3}{(x-1)(y-2)}\right] \sin ^{-1}(y-2) \text { exists, then find } 10 \lim _{x \rightarrow \frac{1}{3}} \frac{f^{-1}(x)}{(3 x-1)} .
\]
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