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:

Image

To find the limit of an indeterminate form of the type Image 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 Image and if Image exists, then

Image

if Image 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 Image the same consequences follow as in case (a). The rules in (a) and (b) both hold for one-sided limits.

(c) If Image exists, then

Image

if Image does not exist, then L’Hôpital’s Rule cannot be applied. (Here the notation “x → ∞” represents either “x → + ∞” or “x → −∞.”)

(d) If Image the same consequences follow as in case (c).

In applying any of the above rules, if we obtain Image 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

Image is of type Image and thus equals Image

(Compare with Example 12 from Chapter 1.)

EXAMPLE 39

Image is of type Image and therefore equals Image

EXAMPLE 40

Image (Example 13) is of type Image and thus equals Image as before. Note that Image is not the limit of an indeterminate form!

EXAMPLE 41

Image is of type Image and therefore equals Image

EXAMPLE 42

Image (Example 20) is of type Image so that it equals Image which is again of type Image Apply L’Hôpital’s Rule twice more:

Image

For this problem, it is easier and faster to apply the Rational Function Theorem!

EXAMPLE 43

Find Image

SOLUTION: Image is of type Image and equals Image

EXAMPLE 44

Find Image

SOLUTION: Image

BEWARE: L’Hôpital’s Rule applies only to indeterminate forms Image Trying to use it in other situations leads to incorrect results, like this:

Image

L’Hôpital’s Rule can be applied also to indeterminate forms of the types 0 · ∞ and ∞ − ∞, if the forms can be transformed to either Image

EXAMPLE 45

Find Image

SOLUTION: Image is of the type ∞ · 0. Since x Image and, as x → ∞, the latter is the indeterminate form Image we see that

Image

(Note the easier solution Image

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 Image

SOLUTION: Image is of type 1. Let y = (1 + x)1/x, so that

ln Image ln (1 + x). Then Image ln Image which is of type Image Thus,

Image

and since Image ln y = 1, Image y = e1 = e.

EXAMPLE 47

Find Image

SOLUTION: Image is of type ∞0. Let y = x1/x, so that ln Image

(which, as x → ∞, is of type Image). Then Image ln Image and Image y = e0 = 1.

For more practice, redo the Practice Exercises, applying L’Hôpital’s Rule wherever possible.