`lim_(xto0) (cos(tanx)-cosx)/(x^(4))` is equal to

Options

A. `f(1+0)=-1,f(1-0)=0`
B. `f(1+0)=0=f(1-0)`
C. `underset(xto1)limf(x)` exists
D. `underset(xto1)f(x)` does not exist

Answer

B

Explanation

`cos(tanx)-cosx=2sin((x+tanx)/(2))sin((x-tanx)/(2))`
or `underset(xto0)lim(cos(tanx)-cosx)/(x^(4))`
`=underset(xto0)lim(2sin((x+tanx)/(2))sin((x-tanx)/(2)))/(x^(4))`
`=underset(xto0)lim(2sin((x+tanx)/(2))sin((x-tanx)/(2)))/(x^(4)((x+tanx)/(2))((x-tanx)/(2)))((x^(2)-tan^(2)x)/(4))`
`=(1)/(2)underset(xto0)lim(x^(2)-tan^(2)x)/(x^(4))`
`=(1)/(2)underset(xto0)lim(x^(2)-(x+(x^(3))/(3)+(2)/(15)x^(5)+...)^(2))/(x^(4))`
`=(1)/(2)underset(xto0)lim(1)/(x^(2))(1-(1+(x^(2))/(3)+(2)/(15)x^(4)+...)^(2))=-(1)/(3)`