TITLE: Evaluating a limit, $\lim\limits_{t\rightarrow 0^{+}} {\sum\limits_{n=1} ^{\infty} \frac{\sqrt{t}}{1+tn^2}}$
QUESTION [7 upvotes]: 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*}

REPLY [12 votes]: Hint: 
$$
\sqrt t \int_1^\infty  {\frac{1}{{1 + tx^2 }}\,dx}  = \int_{\sqrt t }^\infty  {\frac{1}{{1 + x^2 }}\,dx}  \to \frac{\pi }{2}.
$$