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  4. The value of displaystyle lim m arrow ∞( cos (x/m))m is

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Q. The value of $\displaystyle\lim _{m \rightarrow \infty}\left(\cos \frac{x}{m}\right)^{m}$ is

Limits and Derivatives

Solution:

$\displaystyle\lim _{m \rightarrow \infty}\left(\cos \frac{x}{m}\right)^{m}=\displaystyle\lim _{m \rightarrow \infty}\left[1-\left(1-\cos \frac{x}{m}\right)\right]^{m}$
$=\displaystyle\lim _{m \rightarrow \infty}\left[1-2 \sin ^{2} \frac{x}{2 m}\right]^{m}$
$=e^{\displaystyle\lim _{m \rightarrow \infty}-2 \sin ^{2} \frac{x}{2 m} \cdot m}=1$