GATE CS 2019 Q.23 (CS Q.13): Calculus

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

$1$

$53/12$

$108/7$

Limit does not exist

Official answer and solution

Solution

Apply algebraic factorization or L'Hopital's rule to evaluate the limit of the indeterminate 0/0 form.

We need to evaluate:

L = \lim_{x \to 3} \frac{x^4 - 81}{2x^2 - 5x - 3}

Substituting $x = 3$ :

Numerator: $3^4 - 81 = 81 - 81 = 0$
Denominator: $2(3^2) - 5(3) - 3 = 18 - 15 - 3 = 0$
This is an indeterminate form of type $\frac{0}{0}$ .

Applying L'Hopital's rule by differentiating the numerator and denominator with respect to $x$ :

\frac{d}{dx}(x^4 - 81) = 4x^3

\frac{d}{dx}(2x^2 - 5x - 3) = 4x - 5

Evaluating the limit as $x \to 3$ :

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

Final answer

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