`lim_(xto0)(1+x)^(1/(3x))`

Options

A. `logn((2)/(3))`
B. 0
C. `n log n((2)/(3))`
D. not defined

Answer

A

Explanation

`underset(xto1)lim(1+sinpi((3x)/(1+x^(2))))/(1+cospix)=underset(xto1)lim(1-cos"((3pi)/(2)-(3pix)/(1+x^(2))))/(1-cos(pi-pix))`
`=underset(xto1)lim(2sin^(2)"((3pi)/(4)-(3pix)/(2(1+x^(2)))))/(2-sin^(2)((pi)/(2)-(pix)/(2)))`
`=underset(xto1)lim(((3pi)/(4)-(3pix)/(2(1+x^(2))))/((pi)/(2)-(pix)/(2)))^(2)`
`=underset(xto1)9(((1)/(2)-(x)/(1+x^(2)))/(1-x))^(2)`
`=underset(xto1)lim9((x-1)/(2(1+x^(2))))^(2)=0`