The value of $\underset{n\to \mathrm{\infty }}{lim} {\left(\cos \frac{x}{n}\right)}^n$, is 

Solution:

we have,

$\begin{array}{l}\underset{n\to \mathrm{\infty }}{lim} {\left(\cos \frac{x}{n}\right)}^n\\ =\underset{n\to \mathrm{\infty }}{lim} {\left\{1+\left(\cos \frac{x}{n}-1\right)\right\}}^n\\ =\underset{n\to \mathrm{\infty }}{lim} {\left\{1-2{\sin }^2\frac{x}{2n}\right\}}^n\end{array}$

$=e^{\underset{n\to \mathrm{\infty }}{lim} -2n{\sin }^2\frac{x}{2n}}$

$=e^{-\underset{n\to \mathrm{\infty }}{lim} {\left(\frac{\sin \frac{x}{2n}}{x/2n}\right)}^2\times \frac{1}{2n}}$

$=e^{-1\times 0}=e^0=1$