Evaluate: `lim_(xrarr0) (tan x-sin x)/(x^(3))`

Options

A. `1`
B. `2`
C. `1/2`
D. `0`

Answer

C

Explanation

`underset(xrarr0)"lim"(tan x-sin x)/(x^(3))`
`=underset(xrarr0)"lim"((sinx)/(cosx)-sinx)/(x^(3)) `
`underset(xrarr0)"lim"(sin x-cos x sin x)/(x^(3)cos x)`
`=underset(xrarr0)"lim"(sin x(1-cos x))/(x^(3)cos x).((1+cos x))/((1+cos s))`
`=underset(xrarr0)"lim"(sin x(1-cos^(2)x))/(x^(3).cos x (1+cos x))`
`=underset(xrarr0)"lim"(sin^(3)x)/(x^(3).cos x(1+cos x))`
`=underset(xrarr0)"lim"((sin x)/(x))^(3).(1)/(cos x(1+cos x))`
`=1xx(1)/(cos 0(1+cos 0))`
`(because underset(xrarr0)"lim"(sinx)/(x)=1)`
`=(1)/(2)`