For which $a$ and $b$ is $f(x) = \begin{cases} \frac{\sin ax}{x}, & x < 0 \\ 4, & x = 0 \\ \frac{e^{bx} - 1}{2x}, & x > 0 \end{cases}$ continuous at $x = 0$?

Explanation

1. 01

   Left: $\frac{\sin ax}{x} = a \cdot \frac{\sin ax}{ax} \to a$, and it must equal $f(0) = 4$, so $a = 4$.

2. 02

   Right: $\frac{e^{bx} - 1}{2x} = \frac{b}{2} \cdot \frac{e^{bx} - 1}{bx} \to \frac{b}{2}$, so $\frac{b}{2} = 4$ and $b = 8$.