Mathematics

The key is recognizing that for a piecewise function to have a limit at a boundary point, the limit of the formula on one side must match the value (or limit) on the other side. Here, you need the limit of $a\frac{\sin x}{x}$ as $x \to 0$ to equal $q(0) = 3$. Use the fundamental limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$ to find what $a$ must be.

The key is recognizing that for a piecewise function to have a limit at a boundary point, the limit of the formula on one side must match the value (or limit) on the other side. Here, you need the limit of $a\frac{\sin x}{x}$ as $x \to 0$ to equal $q(0) = 3$. Use the fundamental limit $\lim_{x \to 0} \frac{\sin x}{x} = 1$ to find what $a$ must be.