Too Much Calculus May Put You In The L'Hôpital

There are many functions that are revealed to be undefined after an application of L'Hôpital's rule. One way of determining such a function is producing an undefined form (such as 1/0) after deciding a value of $x$ to approach. For instance, the limit $\displaystyle\lim_{x \to 0} \dfrac{e^x}{2x}$ will produce the desired undefined value. We need only take the antiderivative of the numerator and denominator to determine the original indeterminate limit.

$\frac{e^x}{2x} \underbrace{\longrightarrow}_{\text{working backwards}} \frac{e^x + C}{x^2} \longrightarrow \frac{e^x - 1}{x^2} .$

The value of C is chosen so that the limit $\lim_{x \to 0} \dfrac{e^x - 1}{x^2}$ has the indeterminate form 0/0, but when L'Hôpital's rule is applied and the limit is evaluated, the undefined form 1/0 is achieved.