The value of the following limit is ___________.
lim_{x - 0} (ln(1 + x) - x) / x^2

Question

The value of the following limit is ___________.
lim_{x -> 0} (ln(1 + x) - x) / x^2

Accepted Answer

-0.5 Substituting x = 0 gives 0/0. Apply L'Hopital's rule once: differentiate numerator (1/(1+x) - 1 = -x/(1+x)) and denominator (2x). The new limit is (-x/(1+x))/(2x) = -1/(2(1+x)), which at x = 0 equals -1/2 = -0.5. Alternatively, the Taylor expansion ln(1+x) = x - x^2/2 + ... gives numerator ~ -x^2/2, so the limit is -1/2.

The value of the following limit is ___________. lim_{x -> 0} (ln(1 + x) - x) / x^2