In the Limits module of the AP Calculus AB course, we studied limits at infinity as well as limits at zero. However, there were some examples where if we simply plugged in zero or infinity, we would get an indeterminate form. An indeterminate form is either 0/0 or ±∞/±∞ where the ± signifies that there are multiple combinations of ∞ and -∞ that can be a part of an indeterminate form. For the limits at infinity, we found ways to classify the behavior of the function based on the highest power in the numerator and denominator (as well as any oscillating functions). But how do we obtain limits at zero that have an indeterminate form.
L’Hopital’s Rule states that if a limit as an indeterminate form at some point x=a, we have the following equality:
Essentially, taking the derivative of the numerator and denominator independently (without using the Quotient Rule) and then evaluating the limit yields the same value as for the original limit.
Consider two of the special trig limits:
Both of the limits can be verified using L’Hopital’s Rule. Notice that if you plug in 0 to either of these limits, you will get the indeterminate form 0/0. Taking the derivatives of the numerator and denominator of both of these expressions yields a simple expression without x in the denominator. Then, plugging in 0 verifies the two limits.
But what happens if you still get an indeterminate form after you use L’Hopital’s Rule? You can simply differentiate again. L’Hopital’s Rule holds for however many derivatives are necessary to get a limit that is not in an indeterminate form.