Differentiation Question 178
Question: The value of $ \underset{x\to 0}{\mathop{\lim }}{\log _{e}}{{(sinx)}^{\tan x}} $ is
Options:
A) 1
B) -1
C) 0
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ \underset{x\to 0}{\mathop{\lim }}{\log _{e}}{{(sinx)}^{\tan x}}=\underset{x\to 0}{\mathop{\lim }}\tan x.{\log _{e}}\sin x $
$ (0\cdot \infty form) $
$ \underset{x\to 0}{\mathop{\lim }}\frac{{\log _{e}}\sin x}{\cot x}( \frac{\infty }{\infty }form ) $
$ =\underset{x\to 0}{\mathop{\lim }}\frac{\cot x}{-\cos ec^{2}x} $ [Using L-Hospital’s rule] $ =\underset{x\to 0}{\mathop{\lim }}(-cosx.sinx)=0. $
BETA
