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For a very small value of x, so that terms of order x³ and above can be ignored, the expression ((1+x)^(3/2)-(1+(1)/(2)x)³)/(1-x)^(1/2) is approximately equal to

1. 1 - 3/8 x²
2. 3x + 3/8 x²
3. -3/8 x²
4. x/2 - 3/8 x²

Correct answer: -3/8 x²

Solution

(1+x)^(3/2) ~ 1 + 3x/2 + 3x^2/8; (1+x/2)^3 ~ 1 + 3x/2 + 3x^2/4; their difference ~ -3x^2/8. Dividing by (1-x)^(1/2) ~ 1 + x/2 leaves -3/8 x^2 to this order.

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