Question #40968

This question tests the rigorous definition of even multiplicity roots and their interaction with logarithmic series expansions in one-sided limits.

The primary trap is the mechanical application of L'Hopital's rule or Taylor series without respecting the piecewise nature of the square root function.

Students commonly and incorrectly assume $\sqrt{x^2} = x$ regardless of the domain.

If a student blindly substitutes $\sqrt{x^2} = x$, the expression incorrectly simplifies to $\frac{\ln(1+x)-x}{x^2}$, yielding the classic standard limit of $-1/2$.

By forcing a left-hand limit, the problem demands the correct substitution $\sqrt{x^2} = -x$, which drastically alters the linear term inside the logarithm and completely changes the coefficient of the quadratic term in the expansion.