Explanation

(b)
$\lim_{x \rightarrow 0}\frac{e^{ax} - e^{bx}}{x}$
$= \lim_{x \rightarrow 0}\frac{\left( 1 + \frac{ax}{1!} + \frac{(ax)^{2}}{2!} + ... \right) - \left( 1 + \frac{bx}{1!} + \frac{(bx)^{2}}{2!} + ... \right)}{x}$
$= a - b$
Alternate
$\lim_{x \rightarrow 0}\frac{e^{ax} - e^{bx}}{x} = \lim_{x \rightarrow 0}\frac{ae^{ax} - be^{bx}}{1} = a - b$