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Question #40364

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the following limit: $\lim_{x \to 0} \frac{\left[ 256 - x^2 \sin^2\left(\frac{1}{x}\right) \right]}{x^2 + 1}$ where $[ \cdot ]$ denotes the greatest integer function.

Options

A.
256
B.
255
C.
0
D.
\text{Does not exist}

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

Greatest Integer Function (GIF)|Squeeze Theorem (Sandwich Theorem)|Step Function Discontinuities

Why This Question Is Tricky

Common Mistakes

This question tests the dangerous assumption that $\lim [f(x)] = [\lim f(x)]$ when applying the Squeeze Theorem.

Students often correctly use the Squeeze Theorem to find that the inner function approaches 256, leading them to incorrectly choose 256 (or 255 if they assume it strictly approaches from below).

The trap lies in failing to recognize that $x^2 \sin^2(1/x)$ oscillates and touches 0 infinitely many times as $x \to 0$ .

This forces the GIF to jump between 255 and 256, creating jump discontinuities that invalidate standard limit shortcuts and prove the limit does not exist.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Greatest Integer Function (GIF)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free