If `L=lim_(xto0) (1)/(x^(3))((1)/(sqrt(1+x))-(1+ax)/(1+bx))` exists,then

Options

A. `cosLltcosR`
B. `tan(2L)ltanlt2R`
C. `sin LgtsinR`
D. `None of these

Answer

A, B, C

Explanation

`L=underset(xto0)lim(1)/(x^(3))((1)/(sqrt(1+x))-(1+ax)/(1+bx))`
`=underset(xto0)lim(1+bx-(1+ax)sqrt(1+x))/(x^(3))underset(xto0)lim(1)/(sqrt(1+x)(1+bx))`
`=underset(xto0)lim(1+bx-(1+ax)(1+x)^(1//2))/(x^(3))`
`(1+bx-(1+ax)(1+(1)/(2)xx+((1)/(2)((1)/(2)-1))/(2!)x^(2)+((1)/(2)((1)/(2)-1)((1)/(2)-2))/(3!)x^(3)+...))/(x^(3))`
`=underset(xto0)lim(1+bx-(1+ax)(1+(1)/(2)xx-(1)/(8)x^(2)+(1)/(16)x^(3)+.........))/(x^(3))`
`=underset(xto0)lim((b-(1)/(2)-a)x+((-a)/(2)+(1)/(8))x^(2)+((a)/(8)-(1)/(16))x^(3))/(x^(3))`
Limit exists, if `b-(1)/(2)-a=0" and "-(1)/(8)+(a)/(2)=0`
`:." "a=(1)/(4)" and "b=(3)/(4)`
`:." "L=-(1)/(32)`