Continuity — Question 6

Question 6

Let the function $f(x)$ be defined as: $$f(x) = \begin{cases} \frac{\sin(ax)}{x}, & x \neq 0 \\ b, & x = 0 \end{cases}$$

Find the values of constants $a$ and $b$ such that $f(x)$ is continuous at $x = 0$.

Question 6 - Solution

To ensure continuity at $x = 0$, we must have: $$\lim_{x \to 0} f(x) = f(0) = b$$

We are given: $$f(x) = \frac{\sin(ax)}{x}, \quad x \neq 0$$ $$f(0) = b$$

We evaluate the limit: $$\lim_{x \to 0} \frac{\sin(ax)}{x} = \lim_{x \to 0} a \cdot \frac{\sin(ax)}{ax} = a \cdot \lim_{u \to 0} \frac{\sin u}{u} = a \cdot 1 = a$$

So for continuity, we must have: $$\lim_{x \to 0} f(x) = a = b$$

Final Answer: $$\boxed{\text{Any } a \in \mathbb{R} \text{ and } b = a}$$