A numerical analyst studies the error of a three-term approximation of the exponential…

Step 1. First use. The top gives $e^x - 1 - x$ and the bottom gives $3x^2$. At $x = 0$ both vanish, so the form is still $\dfrac{0}{0}$.

Step 2. Second use. The top gives $e^x - 1$ and the bottom gives $6x$. Both vanish again.

Step 3. Third use. The top gives $e^x$ and the bottom gives $6$. This is not indeterminate: $\dfrac{e^x}{6} \to \dfrac{1}{6}$.

Step 4. The answer. The limit is $\dfrac{1}{6} \approx 0.1667$, and the rule was used $3$ times.