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lim x--->0 1-cos(1- cos x )/x^4 ans is 1/2

lim x--->0
1-cos(1- cos x )/x^4 
ans is 1/2

Grade:11

2 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 years ago
Ans: 1/8
Sol:
L = \lim_{x\rightarrow 0}\frac{(1-cos(1-cosx))}{x^{4}}
L = \lim_{x\rightarrow 0}\frac{(1-cos(1-cosx))}{(1-cosx)^{2}}.\frac{(1-cosx)^{2}}{x^{4}}….............(1)
Using the expansion expression of cosx, we have
cosx = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - ..........
I = \lim_{x\rightarrow 0}\frac{1-cosx}{x^{2}} = \frac{1}{2}
Using value of I in (1), we have
L = \lim_{x\rightarrow 0}\frac{1}{2}.\frac{(1-cosx)^{2}}{x^{4}}
L = \frac{1}{2}\lim_{x\rightarrow 0}(\frac{1-cosx}{x^{2}})^{2} = \frac{1}{2}.(\frac{1}{2})^{2}
L = \frac{1}{8}
Cheers!
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty
moidin afsan
20 Points
9 years ago
ans is ½
 

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