If `x` is so small that `x^(3)` and higher power of `x` may neglected, then `((1 + x)^(3/2) - (1 + 1/2x)^(3))/(1 - x)^(1/2)` may be approximated as

Options

A. `3x+3/(x^(2))`
B. `1-3/8 x^(2)`
C. `x/2-3/x x^(2)`
D. `-3/8 x^(2)`

Answer

D

Explanation

`((1-x)^(3//2)(1+(1)/(2)x)^(3))/((1-x)^(1//2))=((1+3/2x+3/8x^(2))-(1+3/2x+3(x^(2))/(4)))/((1-x)^(1//2))`
`= (-3)/(8)x^(2)(1-x)^(-1//2)`
`= - 3/8 x^(2)(1+x/2) = - 3/8 x^(2)`