Functions Question 245
Question: $ \underset{m\to \infty }{\mathop{\lim }}{{( \cos \frac{x}{m} )}^{m}}= $
[AMU 2001]
Options:
A) 0
B) e
C) 1/e
D) 1
Show Answer
Answer:
Correct Answer: D
Solution:
$ \underset{m\to \infty }{\mathop{\lim }},{{( \cos \frac{x}{m} )}^{m}}=\underset{m\to \infty }{\mathop{\lim }},{{[ 1+( \cos \frac{x}{m}-1 ) ]}^{m}} $ $ =\underset{m\to \infty }{\mathop{\lim }},{{[ 1-( -\cos \frac{x}{m}+1 ) ]}^{m}} $ $ =\underset{m\to \infty }{\mathop{\lim }},{{[ 1-2{{\sin }^{2}}\frac{x}{2m} ]}^{m}} $ $ ={e^{\underset{m\to \infty }{\mathop{\lim }},-( 2{{\sin }^{2}}\frac{x}{2m} ),m}} $ $ ={e^{\underset{m\to \infty }{\mathop{\lim }},-2{{( \frac{\sin \frac{x}{2m}}{x/2m} )}^{2}}( \frac{x^{2}}{4m^{2}} ),m}} $ $ ={e^{-2\underset{m\to \infty }{\mathop{\lim }},\frac{x^{2}}{4m}}}=e^{0}=1 $ .
BETA
