Evaluate `lim_(xto0) (e^(x)-e^(-x)-2x)/(x-sinx).`

Answer

2

Explanation

`underset(xto0)lim(e^(x)-e^(-x)-2x)/(x-sinx)`
`=underset(xto0)lim((1+x+(x^(2))/(2!)+(x^(3))/(3!)+...)-(1-x+(x^(2))/(2!)-(x^(3))/(3!)+...)-2x)/(x-(x-(x^(3))/(3!)+(x^(5))/(5!)......))`
`=underset(xto0)lim(2.(x^(3))/(6)+2.(x^(5))/(5!)+...)/((x^(3))/(6)-(x^(5))/(5!)...)`
`=underset(xto0)lim((1)/(3)+(1)/(60)x^(2)+...)/((1)/(6)-(1)/(120)x^(2)+...)=(1//3)/(1//6)=2`