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Question
Prove that `sqrt((1 - x)/(1 + x))` is approximately euqal to `1 - x + x^2/2` when x is very small
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Solution
L.H.S = `sqrt((1 - x)/(1 + x))`
= `(1 - x)^(1/2)/(1 + x)^((-1)/2)`
= `(1 - x)^(1/2) (1 + x)^((-1)/2)`
= `(1 - 1/2 x + (1/2(- 1/2))/(2*1) x^2 ....)(1 - 1/2 x + ((1/2)(3/2))/(2*1) x^2 ....)`
= `(1 - x/2 - x^2/8 ...)(1 - x/2 + (3x^2)/8 ...)`
=`1 - x/2 + (3x^2)/8 - x/2 + x^2/4 - x^2/8 ...`
= `(1 - 2x)/2 + (4x^2)/8 + ....`
= `1 - x + x^2/2`
= R.H.S
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