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the value of lim(x->0)(cos(sinx)-cosx)/x...

the value of `lim_(x->0)(cos(sinx)-cosx)/x^4` is equal to:

A

`1//5`

B

`1//6`

C

`1//4`

D

`1//2`

Text Solution

Verified by Experts

The correct Answer is:
B
`lim_(x to 0)(cos(sinx)-cos x)/(x^(4))`
`=lim_(x to 0) (2 sin((x+sinx)/(2)) sin((x-sinx)/(2)))/(x^(4))`
`=2 lim_(x to 0) [(sin((x+sin x)/(2)))/(((x+ sin x)/(2)))xx(sin((x-sin x)/(2)))/(((x-sin x)/(2))) xx ((x+sin x)/(2x))((x-sinx)/(2x^(3)))]`
`= 2 lim_(x to 0) [(sin((x+sinx)/(2)))/((x+sinx)/(2))xx(sin((x-sinx)/(2)))/((x-sinx)/(2)) xx ((1)/(2)+(sinx)/(2x))((x-sinx)/(2x^(3)))]`
`=2 xx 1xx1xx((1)/(2) +(1)/(2))lim_(x to 0) (x-sinx)/(2x^(3))`
`=lim_(x to 0) (x-sinx)/(x^(3))=lim_(x to 0)(x-(x-(x^(3))/(3!)+(x^(5))/(5!)- ...))/(x^(3))`
`=lim_(x to 0) ((1)/(3!)-(x^(2))/(5!)+...)=(1)/(6)`
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