Mathematics - Limits Question with Solution | TestHub

${\lim }_{n\to \infty }\left[\frac{1}{n^2}{\sec }^2\frac{1}{n^2}+\frac{2}{n^2}{\sec }^2\frac{4}{n^2}+\frac{3}{n^2}{\sec }^2\frac{9}{n^2}+\dots .+\frac{1}{n}{\sec }^{2}1\right]$ is equal to ${\lim }_{n\to \infty }\frac{r}{n^2}{\sec }^2\frac{r^2}{n^2}={\lim }_{n\to \infty }\frac{1}{n}\cdot \frac{r}{n}{\sec }^2\frac{r^2}{n^2}$ $\Rightarrow$ Given limit is equal to value of integral ${\int }_0^{1}x{\sec }^2{x}^{2}dx$ or $\frac{1}{2}{\int }_0^{1}2x\sec x^{2}dx=\frac{1}{2}{\int }_0^1{\sec }^{2}tdt$ [put $x^2=t]$ $=\frac{1}{2}(\tan t{)}_0^1=\frac{1}{2}\tan 1$.