calculus - Evaluating a limit, $limlimits_{trightarrow 0^{+}} {sumlimits_{n=1} ^{infty} frac{sqrt{t}}{1+tn^2}}$

It seems that the following limit exists. But I couldn't figure out the exact value. Anyone could help me? Thanks!
$$\begin{align*} \lim_{t\rightarrow 0^{+}} {\sum_{n=1} ^{\infty} \frac{\sqrt{t}}{1+tn^2}} \end{align*}$$

Answer

Hint:
$$\sqrt t \int_1^\infty {\frac{1}{{1 + tx^2 }}\,dx} = \int_{\sqrt t }^\infty {\frac{1}{{1 + x^2 }}\,dx} \to \frac{\pi }{2}.$$