What is lim x = 0, [ (sqrt (1+ x) - 1 - x/2)/ x^2]
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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
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