Question #41223

This question tests the evaluation of indeterminate forms arising from the determinant of a variable matrix, specifically targeting the $1^\infty$ form.

The trap lies in the 'Denominator Restriction' and limit splitting: students often incorrectly assume $\lim (f(x)g(x))^{h(x)} = \lim f(x)^{h(x)} \lim g(x)^{h(x)}$.

Here, the individual limits $\lim_{t \to 0} (\frac{e^t-1}{t})^{1/t^2}$ and $\lim_{t \to 0} (\frac{\ln(1+t)}{t})^{1/t^2}$ diverge to $\infty$ and $0$ respectively due to the non-zero $O(t)$ terms in their expansions.

However, their product cancels the $O(t)$ term, making the combined limit converge to $e^{1/4}$.

Blindly applying L'Hôpital's rule or splitting the limit leads to a lethal collision of undefined terms.