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Question #40925

Difficulty: 8/10 | 🟢 Free

The Question

Let a curve $C$ be defined parametrically by $x(t) = (t-1)^3$ $y(t) = \lim_{n \to \infty} \frac{t^{2n} (t-1)^3 (t^2 + 4) + |t-1|^3 (t^3 + 1)}{t^{2n} + 1}$ for $t \in (0, \infty)$ . Let $L = \lim_{x \to 0^+} \frac{dy}{dx}$ and $R = \lim_{x \to 0^-} \frac{dy}{dx}$ . Find the value of $10L + R$ .

Options

A.
52
B.
48
C.
-22
D.
55

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

Indeterminate Forms|Polynomial Continuity/Differentiability Link|Modulus Sign Dependency|Parametric Derivative Endpoint Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Parametric Derivative Endpoint Trap' combined with 'Modulus Sign Dependency' in limit-defined functions.

Students often blindly apply L'Hopital's rule or differentiate the parametric equations without analyzing the piecewise nature of $y(t)$ caused by the $\lim_{n \to \infty} t^{2n}$ term.

Furthermore, the modulus $|t-1|^3$ requires careful sign evaluation for $t < 1$ .

Failing to recognize that $x \to 0^-$ corresponds to $t \to 1^-$ and dropping the negative sign from the modulus leads to the incorrect right-hand derivative, trapping students who rely on rote differentiation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Indeterminate Forms
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free