\begin{align} \lim_{x \to 0} \frac{b^x - c^x}{ax} &= \lim_{x \to 0} \frac{e^{x \log b} - e^{x \log c}}{ax} \\ &= \lim_{x \to 0} \frac{\log b \cdot e^{x \log b} - \log c \cdot e^{x \log c}}{a} \\ &= \frac{\log b - \log c}{a} \\ &= \frac{1}{a} \log \frac{b}{c} \end{align}