$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {\frac{r}{{{n^2}}}{{\sec }^2}\frac{{{r^2}}}{{{n^2}}}} $
$= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{r = 1}^n {\frac{r}{n}{{\sec }^2}\frac{{{r^2}}}{{{n^2}}}} $
Given limit is equal to the value of integral $\int_0^1 {x{{\sec }^2}{x^2}dx} $
=$\frac{1}{2}\int_0^1 {2x{{\sec }^2}{x^2}dx} = \frac{1}{2}\int_0^1 {{{\sec }^2}t\;dt} $, [Put ${x^2} = t$]
$ = \frac{1}{2}[\tan \;t]_0^1 = \frac{1}{2}\tan 1$.
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