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Question #41494

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the limit: $\lim_{x \to 0} \frac{1}{x^4} \int_{\ln(1+x)}^{x} \frac{t^2 e^t}{1+t^2} \, dt$

Options

A.
$\frac{1}{2}$
B.
$\frac{1}{4}$
C.
$\frac{1}{3}$
D.
1

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Integration by Parts (LIATE Rule)|Newton-Leibniz Formula with variable limits|Indeterminate Forms|L'Hopital's Pitfall

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Newton-Leibniz formula in indeterminate forms, specifically targeting the 'L'Hopital's Pitfall' where students neglect the chain rule on variable limits.

When differentiating the lower limit $\ln(1+x)$ , the required factor $\frac{1}{1+x}$ perfectly cancels the $e^{\ln(1+x)}$ term from the integrand.

If a student forgets this chain rule differentiation, the uncancelled $(1+x)$ alters the $x^3$ coefficient in the Taylor expansion (yielding $x^2$ instead of $x^2-x^3$ ), leading directly to the trap answer $1/4$ .

It also rewards students who recognize that the leading order behavior can be extracted by approximating the integrand and limits directly.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Integration by Parts (LIATE Rule)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free