A chemist compares two rates built from exponentials…

A chemist compares two rates built from exponentials. The first rate is $P(x) = \dfrac{e^x - e^{-x} - 2x}{x^3}$ and the second is $S(x) = \dfrac{e^{2x} - 1 - 2x}{x^2}$, for $x \ne 0$ in suitable units. Both have a top and a bottom that vanish at $x = 0$. Use the series $e^t = 1 + t + \dfrac{t^2}{2} + \dfrac{t^3}{6} + \dots$ to find the limit of $P(x)$ as $x \to 0$, to four decimal places, and the limit of $S(x)$ as $x \to 0$.