BC ONLYLimits of the following forms are called indeterminate:
To find the limit of an indeterminate form of the type we apply L’Hôpital’s Rule, which
involves taking derivatives of the functions in the numerator and denominator. In the following, a is a
finite number. The rule has several parts:
(a) If and if exists†, then
if does not exist, then L’Hôpital’s Rule cannot be applied.
- Although this a required topic only for BC students, AB students will find L’Hôpital’s Rule very helpful.
† The limit can be finite or infinite (+∞ or −∞).
(b) If the same consequences follow as in case (a). The rules in (a) and (b)
both hold for one-sided limits.
(c) If exists, then
i f does not exist, then L’Hôpital’s Rule cannot be applied. (Here the notation “x → ∞”
represents either “x → + ∞” or “x → −∞.”)
(d) If the same consequences follow as in case (c).
In applying any of the above rules, if we obtain again, we can apply the rule once more,
repeating the process until the form we obtain is no longer indeterminate.
Examples 38–43 are BC ONLY.
EXAMPLE 38
is of type and thus equals
(Compare with Example 12 from Chapter 1.)EXAMPLE 39
is of type and therefore equals