GATE CS 2016 Set 1 Q.13 (CS Q.3): Calculus

The value of

\lim_{x \to 4} \frac{\sin(x - 4)}{x - 4}

is _____.

Your answer

Official answer and solution

Solution

Substitute $u = x - 4$ to evaluate the standard limit $\lim_{u \to 0} \frac{\sin u}{u}$ .

We evaluate the limit:

L = \lim_{x \to 4} \frac{\sin(x - 4)}{x - 4}

Let $u = x - 4$ . As $x \to 4$ , $u \to 0$ . Substituting $u$ :

L = \lim_{u \to 0} \frac{\sin u}{u}

From fundamental calculus (or applying L'Hopital's rule on the $0/0$ indeterminate form):

\lim_{u \to 0} \frac{\sin u}{u} = \lim_{u \to 0} \frac{\cos u}{1} = \cos(0) = 1

Final answer

Common trap

Evaluating $\sin(0)/0$ as undefined without applying standard limits or L'Hopital's rule.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.