Apply L'Hopital's Rule to the Indeterminate Forms "∞ - ∞" and "∞ * 0"

1. The Indeterminate Form ∞ – ∞

The form occurs when there is a difference between two expressions that are both tending toward as .

EXAMPLE

Evaluate the following limit:

Since and as , we have a limit of the form . One strategy is to write it as a single fraction, since this is a more familiar scenario.

Since , we have the following:

	Start with the limit that needs to be evaluated.
	Replace the expression with a single fraction.
	As x approaches 0 from the right, x - 1 approaches -1 and is a small positive number.
	A negative number divided by a small positive number is a large negative number.

Thus, .

big idea

Many might think that a limit of the form should be 0 since you are “subtracting something from itself.” As we can see, this is not the case. Once we see produce another value, we will see why it is an indeterminate form.

watch

In this video, we'll evaluate .

Considering the results from these last two examples, it is clear now why is an indeterminate form. In one case, the result was , and in another case, the result was .

2. The Indeterminate Form ∞ ᐧ 0

The indeterminate form is handled in one of two ways.

Loosely speaking, we can say that a limit of the form will approach 0 and a limit of the form will approach .

That said, we can treat “0” and “” as reciprocals as far as limits are concerned.

This means that the indeterminate form could be rewritten as either or , whichever is more convenient.

EXAMPLE

Evaluate the following limit:

If we look at each factor separately, we see that and as . Thus, this limit has the form

To rewrite, consider the fact that , which means , which now has the form .

To evaluate, use L’Hopital’s rule.

	Start with the limit that needs to be evaluated.
	Since and are differentiable and the limit has the form , L’Hopital’s rule is used.
	Remove the common factor of 2.
	Since has the form , continue to use L’Hopital’s rule.
	Since the denominator grows very large as , the limit is 0.

Thus, .

big idea

If has the form , write or , then use L'Hopital's rule.

watch

In this video, we’ll evaluate .

watch

In this video, we’ll evaluate .

summary

In this lesson, you learned that with the addition of new indeterminate forms, more strategies need to be used. Specifically, you learned that for the indeterminate form , combining the fractions or rationalizing are the most common strategies; for the indeterminate form , rewriting the expression using reciprocals then using L’Hopital’s rule is the main strategy.