Question

$\underset{n\to \infty }{\lim }{\left(\cos \frac{x}{n}\right)}^n$ is equal to

• A. $e^1$
• B. $e^{-1}$
• C. $1$
• D. None of these

Answer 1

Answer: (C)

$\underset{n\to \infty }{\lim }{\left(\cos \frac{x}{n}\right)}^n=e^{\underset{n\to \infty }{\lim }n\left(\cos \frac{x}{n}-1\right)}$
$=e^{\underset{n\to \infty }{\lim }-n\cdot 2{\sin }^2\left(\frac{x}{2n}\right)}$
$=e^{-2\underset{n\to \infty }{\lim }{\left(\frac{\sin \left(\frac{x}{2n}\right)}{\frac{x}{2n}}\right)}^2\cdot \frac{x^2}{4n^2}\cdot n}$
$=e^{-2\times \underset{n\to \infty }{\lim }\frac{x^2}{4n}}=e^0=1$