1.

The value of lim_(x to 0) (cos(sin x)-cos x)/(x^(4)) is equal to

Answer»

`1//5`
`1//6`
`1//4`
`1//2`

Solution :`underset(x to 0)LIM (COS (sin x)-cos x)/(x^(4))`
`underset(x to 0)lim (2sin((x+sinx)/(2))sin((x-sinx)/(2)))/(x^(4))`
`2underset(x to 0)lim [(sin ((x+sinx)/(2)))/(((x+sin x)/(2)))xx(sin((x-sinx)/2))/(((x-sin x)/(2)))xx((x +sin x)/(2x))((x-sinx)/(2x^(3)))]`
`2underset(x to 0)lim [(sin ((x+sinx)/(2)))/((x+sin x)/(2))xx(sin((x-sinx)/2))/((x-sin x)/(2))xx((1)/(2)+(sin x)/(2x))((x-sinx)/(2x^(3)))]`
`=2xx1xx1xx((1)/(2)+(1)/(2)) underset(x to 0)lim (x-sinx)/(2x^(3))`
`underset(x to 0)lim (x-sin x)/(x^(3))=underset(x to 0)lim (x-(x-(x^(3))/(3!)+(x^(5))/(5!)-.....))/(x^(3))`
`=underset(x to 0)lim ((1)/(3!)-(x^(2))/(5!)+.....)=(1)/(6)`


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