Question #41882

This question tests the precise conditions for L'Hôpital's Rule and the Fundamental Theorem of Calculus.

The trap baits students into blindly applying L'Hôpital's Rule by differentiating the numerator using Leibniz's rule, which yields $2x \cos(1/x) + 2(e^{3x} - 1)$.

Dividing by the derivative of the denominator ($2x$) gives $\cos(1/x) + \frac{e^{3x}-1}{x}$.

As $x \to 0^+$, the term $\cos(1/x)$ oscillates, leading students to incorrectly conclude that the limit 'Does not exist'.

However, L'Hôpital's Rule only states that if the limit of the derivatives exists, the original limit equals it; the converse is false.

By splitting the integral and using substitution with integration by parts, we reveal that the highly oscillatory term's integral decays faster than $x^2$, yielding a well-defined finite limit.