The value of $\underset{x\to \frac{\pi }{2}}{\lim }{\left(1^{\frac{1}{{\cos }^{2}x}}+2^{\frac{1}{{\cos }^{2}x}}+.\dots \dots \dots +{10}^{\frac{1}{{\cos }^{2}x}}\right)}^{{\cos }^{2}x}$ is equal to

Let, $L=\underset{x\to \frac{\pi }{2}}{\lim }{\left(1^{\frac{1}{{\cos }^{2}x}}+2^{\frac{1}{{\cos }^{2}x}}+.\dots \dots \dots +{10}^{\frac{1}{{\cos }^{2}x}}\right)}^{{\cos }^{2}x}$

Putting $\frac{1}{{\cos }^{2}x}=y$, we get,
$L=\underset{y\to \infty }{\lim }{\left(1^y+2^y+.\dots \dots .+{10}^y\right)}^{\frac{1}{y}}$ $$
$=\underset{y\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{}10{\left[{\left(\frac{1}{10}\right)}^y+{\left(\frac{2}{10}\right)}^y+.\dots \dots .+{\left(\frac{10}{10}\right)}^y\right]}^{\frac{1}{y}}$
$=10{\left(0+0+.\dots \dots +1\right)}^0=10$