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Question #40935

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the following limit: $\lim_{x \to 0} \left( \frac{1}{x^2} \int_0^{x^2} \frac{1+2t^2}{\sqrt{1+t^2}} \, dt \right)^{\csc^4 x}$

Options

A.
$\sqrt{e}$
B.
$e$
C.
$e^{1/3}$
D.
$1$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

The 1^∞ Form Shortcut|Integrating f(x) + xf'(x) forms|Fundamental Theorem of Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the synergy between the $1^\infty$ limit form and the Fundamental Theorem of Calculus (Leibniz Rule).

The primary trap lies in the evaluation of the base limit.

Students who blindly apply L'Hopital's rule to the $1^\infty$ exponent expression $\frac{\int_0^{x^2} ... dt - x^2}{x^2 \sin^4 x}$ often fall victim to the chain rule trap in FTC, forgetting to multiply by the derivative of the upper limit ( $2x$ ).

Alternatively, recognizing the integrand as the exact derivative of $t f(t)$ bypasses the complex differentiation entirely, rewarding algebraic foresight over brute-force calculus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	The 1^∞ Form Shortcut
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free