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Question #41867

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the limit: $\lim_{x \to 2} \frac{\left|\sin\left(\frac{\pi x}{2}\right)\right| + 2\pi \left[ \frac{x}{2} \right] - \pi x \left[ \frac{x}{2} \right]}{2-x}$ where $[t]$ denotes the greatest integer less than or equal to $t$ .

Options

A.
$\frac{\pi}{2}$
B.
$\frac{3\pi}{2}$
C.
$\pi$
D.
\text{Does not exist}

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

Modulus Sign Dependency|Trigonometric Standard Limits|Integral Part and Fractional Part

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous evaluation of limits involving piecewise functions, specifically the modulus and greatest integer function (GIF), at points of potential discontinuity.

A common trap is to blindly apply L'Hôpital's rule by substituting $[x/2] = 1$ and ignoring the modulus, which incorrectly yields $3\pi/2$ .

Another trap is partially applying directional limits but failing to resolve the modulus sign correctly, leading to mismatched LHL and RHL and the false conclusion that the limit does not exist.

The problem is architected such that the jump discontinuity in the GIF perfectly neutralizes the non-differentiable corner of the modulus function, rewarding only students who perform a precise $\epsilon$ -neighborhood analysis for both the left and right limits.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Modulus Sign Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free