Which of the following is/are correct?

Options

A. always 1
B. always -1
C. `(-1)^(m+1)`
D. `(-1)^(n-m)`

Answer

A, B, C

Explanation

`(1)underset(xto0^(+))lim([x+|x|])/(x)=underset(xto0^(+))lim([x+x])/(x)=underset(xto0^(+))lim([2x])/(x)=0`
`underset(xto0^(-))lim([x+|x|])/(x)=underset(xto0^(-))lim([x-x])/(x)=underset(xto0^(-))lim(0)/(x)=0`
`(2)underset(xto0^(+))lim(xe^((1)/(x)))/(1+e^((1)/(x)))=underset(xto0^(+))lim(x)/(1+e^(-(1)/(x)))=0`
`underset(xto0^(-))lim(xe^((1)/(x)))/(1+e^((1)/(x)))=0`
`(3)underset(xto3^(+))lim(x-3)^((1)/(5))sgn(x-3)=underset(xto3^(+))lim(x-3)^((1)/(5))xx1=0`
`underset(xto3^(-))lim(x-3)^((1)/(5))sgn(x-3)=underset(xto3^(-))lim(x-3)^((1)/(5))xx(-1)=0`
`(4)underset(xto0^(+))lim(tan^(-1)|x|)/(x)=underset(xto0^(+))lim(tan^(-1)x)/(x)=1`
`underset(xto0^(-))lim(tan^(-1)|x|)/(x)=underset(xto0^(+))limtan^(-1)(-x)/(x)=-1`