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

moidin afsan

12 Years agoGrade 11

2 Answers

Jitender Singh

12 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

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