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  • Find <img alt="\mathrm{ \lim _{n \rightarrow \infty}\left(\cos \frac{x}{n}\right)^{n^2} }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20%5Clim%20_%7Bn%20%5Crightarrow%20%5Cin

Find \mathrm{ \lim _{n \rightarrow \infty}\left(\cos \frac{x}{n}\right)^{n^2} }
 

Option: 1

\mathrm{e^{-\frac{x^2}{2}}}
 


Option: 2

\mathrm{e^{-\frac{x^2}{4}}}
 


Option: 3

\mathrm{e^{-\frac{x^2}{6}}}
 


Option: 4

None


Answers (1)

best_answer

Assume \mathrm{x} is not a multiple of \mathrm{\pi.} Taylor expand in \mathrm{\frac{x}{n}}:

\mathrm{ \cos \frac{x}{n}=\left(1-\frac{x^2}{2 n^2}+O\left(n^{-4}\right)\right) }

Then

\mathrm{ \lim _{n \rightarrow \infty}\left(\cos \frac{x}{n}\right)^{n^2}=\lim _{n \rightarrow \infty}\left(1-\frac{x^2}{2 n^2}\right)^{n^2}=\lim _{n \rightarrow \infty}\left(1-\frac{x^2}{2 m}\right)^m=e^{-x^2 / 2} }

However, if \mathrm{x=2 k \pi} then the limit is 1 since for all integer \mathrm{n} the value is 1 . And for \mathrm{x=(2 k+1) \pi} the limit does not exist.

Hence option 1 is correct.

 

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