GATE CS 2016 Set 1 Q3 — Calculus: Limits, Continuity, Differentiability

To evaluate the limit:

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

Let us perform a substitution. Let $y = x - 4$.

As $x \to 4$, we have $y \to 0$. Substituting these into the limit expression gives:

$$\lim_{y \to 0} \frac{\sin(y)}{y}$$

Using the standard trigonometric limit identity:

$$\lim_{\theta \to 0} \frac{\sin(\theta)}{\theta} = 1$$

We find that:

$$\lim_{y \to 0} \frac{\sin(y)}{y} = 1$$

Thus, the value of the limit is 1.