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Question #41301

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{|(x-2)(x-4)(x-a)|}{(x-2)(x-4)(x-a)} \sum_{k=1}^5 \frac{k^2 - a^2}{x-k}$ . Find the sum of all real values of $a$ for which $\lim_{x \to a} f(x)$ exists.

Options

A.
3
B.
6
C.
-3
D.
15

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

Limit Existence at Discontinuity|Partial Fractions Decomposition|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the existence of limits at points of discontinuity created by the signum function (Wavy Curve method) interacting with a sum of partial fractions.

Students often fall into two traps: First, they incorrectly assume that $\text{sgn}(P(x))$ always introduces a non-removable discontinuity at its roots, failing to realize that if a root has an even multiplicity (e.g., when $a=2$ or $a=4$ ), the sign does not change, and the limit exists.

Second, they might only check the roots of the denominator of the partial fractions, forgetting to evaluate the general case $a \notin \{1,2,3,4,5\}$ , which yields the hidden valid parameter $a=-3$ where the partial fraction sum evaluates to zero.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Limit Existence at Discontinuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free