Differentiation Question 30
Question: The value of $ \underset{m\to \infty }{\mathop{\lim }}{{( \cos \frac{x}{m} )}^{m}} $ is
Options:
A) 1
B) $ e $
C) $ {e^{-1}} $
D) none of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ \underset{m\to \infty }{\mathop{\lim }}{{( \cos \frac{x}{m} )}^{m}}=\underset{m\to \infty }{\mathop{\lim }}{{[ 1-( 1-\cos \frac{x}{m} ) ]}^{m}} $
$ \underset{m\to \infty }{\mathop{\lim }}{{[ 1-2{{\sin }^{2}}\frac{x}{2m} ]}^{m}} $
$ =\underset{{e^{m\to \infty }}}{\mathop{\lim }}( -2{{\sin }^{2}}\frac{x}{2m} )m=1 $
BETA
