Compute limx → 3 (x4-81)/(2x2-5x-3) A) 1 B) 53/12 C) 108/7 D) Limit does not exist

GATE CSE 2019 · Q13

MCQ1 markEasy

Time00:00

Compute $\lim_{x \to 3} \frac{x^4-81}{2x^2-5x-3}$

Select one answerSingle correct

Pick an option to check

Solution

Official answer: C) 108/7

Calculus

Evaluating the expression at $x = 3$ :

This is an indeterminate form of $\frac{0}{0}$ .

Method 1: Factorization Factor the numerator and denominator:

x^4 - 81 = (x^2 - 9)(x^2 + 9) = (x - 3)(x + 3)(x^2 + 9)

2x^2 - 5x - 3 = 2x^2 - 6x + x - 3 = (2x + 1)(x - 3)

Cancel the common factor $(x - 3)$ for $x \neq 3$ :

\lim_{x \to 3} \frac{(x - 3)(x + 3)(x^2 + 9)}{(2x + 1)(x - 3)} = \lim_{x \to 3} \frac{(x + 3)(x^2 + 9)}{2x + 1}

Substitute $x = 3$ :

\frac{(3 + 3)(3^2 + 9)}{2(3) + 1} = \frac{6 \times 18}{7} = \frac{108}{7}

Method 2: L'Hôpital's Rule Differentiating numerator and denominator with respect to $x$ :

\lim_{x \to 3} \frac{4x^3}{4x - 5} = \frac{4(3^3)}{4(3) - 5} = \frac{4 \times 27}{12 - 5} = \frac{108}{7}

Hence, the correct option is (C).

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This Calculus previous year question is Q13 of the GATE CSE 2019 paper. Find more GATE CSE PYQs on Calculus in Engineering Mathematics, or browse all GATE CSE PYQs.

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Calculus has 40 GATE CSE PYQs across 25 papers from 2011 to 2026. 14 of them are 1-mark MCQs like this one, and it is the only one in GATE CSE 2019.

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.