Functions Question 65
Question: The integer n for which $ \underset{x\to 0}{\mathop{\lim }}\frac{(\cos x-1),(\cos x-e^{x})}{x^{n}} $ is a finite non-zero number is
[IIT Screening 2002]
Options:
A) 1
B) 2
C) 3
D) 4
Show Answer
Answer:
Correct Answer: C
Solution:
n cannot be negative integer for then the limit = 0
Limit $ =\underset{x\to 0}{\mathop{\lim }},\frac{2{{\sin }^{2}}\frac{x}{2}}{2^{2}{{(x/2)}^{2}}}\frac{e^{x}-\cos x}{{x^{n-2}}}=\frac{1}{2}\underset{x\to 0}{\mathop{\lim }},\frac{e^{x}-\cos x}{{x^{n-2}}} $
$ (n\ne 1 $ for then the limit $ =0) $
$ =\frac{1}{2}\underset{x\to 0}{\mathop{\lim }},\frac{e^{x}+\sin x}{(n-2){x^{n-3}}} $ .
So, if $ n=3, $ the limit is $ \frac{1}{2(n-2)} $ which is finite.
If $ n=4, $ the limit is infinite.
BETA
