The value of `underset(xto0)lim(1+sinx-cosx+log(1-x))/(x^(3))` is

Options

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

Answer

C

Explanation

Using expansions of `sinx, cos x and log_e(1-x)` we have
` lim_(xto0) (1+sinx-cosx+log_e(1-x))/(x^3)`
`lim_(xto0)(1+(x-(x^3)/(3 ᴉ)+(x^5)/(5 ᴉ)-(x^7)/(7 ᴉ)+......)-(1-(x^2)/(2 ᴉ )+(x^4)/(4 ᴉ)-(x^6)/(6 ᴉ)+....)-(x+(x^2)/(2)+(x^3)/(3)+(x^4)/(4)....))/(x^3)`
` =lim_(xto0) (-x^3((1)/(3ᴉ)+(1)/(3))-x^4((1)/(4 ᴉ)+(1)/(4))+....)/(x^3)`
`lim_(xto0) -((1)/(3ᴉ)+(1)/(3))-x((1)/(4ᴉ)+(1)/(4))+....=-(1)/(2)`