For each positive integer \(n\),consider the point \(P\),with \(\displaystyle{x}\)-coordinate \(n\) on the curve \(\displaystyle{y^2 - x^2 = 1}\).If \(\displaystyle{d_n}\) represents the shortest distance from point **P** to the line \(\displaystyle{y=x}\),then find \( \displaystyle \lim_{n \to \infty } (n \cdot d_n)\).

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