## 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 0^{0}, 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* = *e*^{1} = *e*.

**EXAMPLE 47**

Find

**SOLUTION:** is of type ∞^{0}. Let *y* = *x*^{1/}* ^{x}*, so that ln

(which, as *x* → ∞, is of type ). Then ln and *y* = *e*^{0} = 1.

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