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Question #40548

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \lim_{n \to \infty} e^{-n x^2}$ . Let $g: \mathbb{R} \to \mathbb{R}$ be a function satisfying the inequality $\cos x - x^2 - 3 f(x) \le g(x) \le \cos x + x^2 - 3 f(x)$ for all $x \in \mathbb{R}$ . If $L = \lim_{x \to 0} g(x)$ and $V = g(0)$ , then the ordered pair $(L, V)$ is:

Options

A.
$(1, -2)$
B.
$(-2, -2)$
C.
$(1, 1)$
D.
$(1, 4)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Closed vs Open Intervals|Squeeze Theorem (Sandwich Theorem)|Limit Existence at Discontinuity

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Squeeze Theorem in the presence of a pointwise convergent sequence of functions that introduces a removable discontinuity.

The trap lies in the assumption that the limit of a function bounded on a closed set (here, $\mathbb{R}$ ) is simply the value of the bounds evaluated at the limit point.

Students often incorrectly substitute $x=0$ directly into the bounds to find the limit, yielding $L = -2$ , failing to recognize that the limit definition relies on the open, punctured neighborhood $x \neq 0$ where the sequence term vanishes.

This highlights the clash between the open boundary nature of limits and the closed evaluation of functions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Inequalities - Class 11
Concept	Closed vs Open Intervals
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free