We have to find lim x-->0 [(sqrt (1+ x) - 1 - x/2)/ x^2]

Replacing x with 0, we see that the expression is of the form 0/0, or indeterminate. This allows us to use L'Hopital's Rule and derive the value of the limit by differentiating the denominator and the numerator.

lim x-->0 [(sqrt (1+ x) - 1 - x/2)/ x^2]

=> lim x-->0 [((1/2)*(1+ x)^(-1/2) - 1/2)/ 2x]

If we replace x with 0, this is again of the form 0/0, so we can differentiate the numerator and denominator again

=> lim x-->0 [ (-(1/4)(1 + x)^(-3/2)/2]

substituting x with 0 we get

(-(1/4)(1 + 0)^(-3/2)/2

=> [-(1/4)] / 2

=> [(-1/4)/2]

=> -1/8

The required value of the limit is -1/8