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# lim x--->01-cos(1- cos x )/x^4 ans is 1/2

Jitender Singh IIT Delhi
7 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