Limit of a point on a curve

Consider the point $\displaystyle{ P(n,\sqrt{n^2 +1}) }$ on the curve.Let the vertical line $\displaystyle{x=n}$ and $\displaystyle{y=x}$ intersect at $\displaystyle{Q(n,n)}$ . Now $\Delta ONQ \sim \Delta PMQ$ . $N \rightarrow \mathcal{foot \quad of \perp from \quad P \quad on} x-axis$ $M \rightarrow \mathcal{foot \quad of \perp from \quad P \quad on} y=x$ $\Rightarrow \dfrac{d_n}{n} = \dfrac{\sqrt{n^2 +1} - n}{n \sqrt{2}}$ So $\lim_{n \to \infty} (n \cdot d_n)=\lim_{n \to \infty} \dfrac{1}{\sqrt{2}} \times \Bigg( 1+ \sqrt{1+ \dfrac{1}{n^2}}\Bigg) = \dfrac{1}{2 \sqrt{2}} = \boxed{0.354}$