Continuity — Question 8

Question 8

Let $$f(x) = \begin{cases} \frac{\sin(3x)}{x}, & x \neq 0 \\ c, & x = 0 \end{cases}$$

Find the value of $c$ that makes $f(x)$ continuous at $x = 0$.

Question 8 - Solution

To ensure continuity at $x = 0$, we require: $$\lim_{x \to 0} f(x) = f(0) = c$$

Compute the limit: $$\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{x \to 0} \frac{3\sin(3x)}{3x} = 3 \cdot \lim_{x \to 0} \frac{\sin(3x)}{3x} = 3 \cdot 1 = 3$$

Thus, to make $f$ continuous at $x = 0$, we must set: $$\boxed{c = 3}$$