Differentiation Question 171
Question: $ \underset{x\to 0}{\mathop{\lim }}[ \cos ec^{3}x.\cot x-2{{\cot }^{3}}x.\cos ecx+\frac{{{\cot }^{4}}x}{\sec x} ] $ is equal to
Options:
A) 1
B) -1
C) 0
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ \underset{h\to 0}{\mathop{\lim }}[ \cos ec^{3}x.\cot x-2{{\cot }^{3}}x.\cos ecx+\frac{{{\cot }^{4}}x}{\sec x} ] $
$ =\underset{x\to 0}{\mathop{\lim }}( \frac{\cos x}{{{\sin }^{4}}x}-\frac{2{{\cos }^{3}}x}{{{\sin }^{4}}x}+\frac{{{\cos }^{5}}x}{{{\sin }^{4}}x} ) $
$ =\underset{x\to 0}{\mathop{\lim }}\frac{\cos x{{(1-cos^{2}x)}^{2}}}{{{\sin }^{4}}x}=\underset{x\to 0}{\mathop{\lim }}\cos x=1. $
BETA
