Calculus AB and Calculus BC
CHAPTER 3 Differentiation
J.* INDETERMINATE FORMS AND L’HÔPITAL’S RULE
BC ONLY
Limits 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
if 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
EXAMPLE 40
(Example 13) is of type
and thus equals
as before. Note that
is not the limit of an indeterminate form!
EXAMPLE 41
is of type
and therefore equals
EXAMPLE 42
(Example 20) is of type
so that it equals
which is again of type
Apply L’Hôpital’s Rule twice more:
For this problem, it is easier and faster to apply the Rational Function Theorem!
EXAMPLE 43
Find
SOLUTION: is of type
and equals
EXAMPLE 44
Find
SOLUTION:
BEWARE: L’Hôpital’s Rule applies only to indeterminate forms Trying to use it in other situations leads to incorrect results, like this:
L’Hôpital’s Rule can be applied also to indeterminate forms of the types 0 · ∞ and ∞ − ∞, if the forms can be transformed to either
EXAMPLE 45
Find
SOLUTION: is of the type ∞ · 0. Since x
and, as x → ∞, the latter is the indeterminate form
we see that
(Note the easier solution
Other indeterminate forms, such as 00, 1∞ and ∞0, may be resolved by taking the natural logarithm and then applying L’Hôpital’s Rule.
BC ONLY
EXAMPLE 46
Find
SOLUTION: is of type 1∞. Let y = (1 + x)1/x, so that
ln ln (1 + x). Then
ln
which is of type
Thus,
and since ln y = 1,
y = e1 = e.
EXAMPLE 47
Find
SOLUTION: is of type ∞0. Let y = x1/x, so that ln
(which, as x → ∞, is of type ). Then
ln
and
y = e0 = 1.
For more practice, redo the Practice Exercises, applying L’Hôpital’s Rule wherever possible.