1. 4.4 Indeterminate Forms and L’Hospital’s Rule
APPLICATIONS OF DIFFERENTIATION
4.4 Indeterminate Forms and L’Hospital’s Rule
CHECK LATER.
In this section, we will learn:
How to evaluate limits of functions whose values cannot be found at certain points.

2. We would like to know the value of the limit
INDETERMINATE FORMS
Expression 1
We would like to know the value of the limit
In computing this limit, we can’t apply Law 5 of limits (Section 2.3) because the limit of the denominator is 0.

3. INDETERMINATE FORM —TYPE 0/0
In general, if we have a limit of the form
where both f(x) → 0 and g(x) → 0 as x → a, then this limit may or may not exist.
It is called an indeterminate form of type 0/0.

4. INDETERMINATE FORMS
Expression 2
Another situation in which a limit is not obvious occurs when we look for a horizontal asymptote of f and need to evaluate the limit
It isn’t obvious how to evaluate this limit because both numerator and denominator become large as x → ∞.

5. There is a struggle between the two.
INDETERMINATE FORMS
There is a struggle between the two.
If the numerator wins, the limit will be ∞.
If the denominator wins, the answer will be 0.
Alternatively, there may be some compromise— the answer may be some finite positive number.

6. INDETERMINATE FORM —TYPE ∞/∞
In general, if we have a limit of the form
where both f(x) → ∞ (or -∞) and g(x) → ∞ (or -∞), then the limit may or may not exist.
It is called an indeterminate form of type ∞/∞.

7. L’HOSPITAL’S RULE
Suppose f and g are differentiable and g'(x) ≠ 0 on an open interval I that contains a (except possibly at a).
Suppose
or that
In other words, we have an indeterminate form of type or ∞/∞.

8. if the limit on the right exists (or is ∞ or - ∞).
L’HOSPITAL’S RULE
Then,
if the limit on the right exists (or is ∞
or - ∞).

9. NOTE 2
The rule is also valid for one-sided limits and for limits at infinity or negative infinity.
That is, ‘x → a’ can be replaced by any of the symbols x → a+, x → a-, x → ∞, or x → - ∞.

10. Find and Thus, we can apply L’Hospital’s Rule: L’HOSPITAL’S RULE
Example 1
Find
and
Thus, we can apply L’Hospital’s Rule:

11. Calculate We have and So, L’Hospital’s Rule gives: L’HOSPITAL’S RULE
Example 2
Calculate
We have and
So, L’Hospital’s Rule gives:

12. However, a second application of l’Hospital’s Rule gives:
Example 2
As ex → ∞ and 2x → ∞ as x → ∞, the limit on the right side is also indeterminate.
However, a second application of l’Hospital’s Rule gives:

13. Calculate As ln x → ∞ and as x → ∞, L’Hospital’s Rule applies:
Example 3
Calculate
As ln x → ∞ and as x → ∞, L’Hospital’s Rule applies:
Notice that the limit on the right side is now indeterminate of type .

14. L’HOSPITAL’S RULE
Example 3
However, instead of applying the rule a second time as we did in Example 2, we simplify the expression and see that a second application is unnecessary:

15. INDETERMINATE FORM—TYPE 0 . ∞
Another kind of limit is called an
indeterminate form of type 0 . ∞.
We can deal with it by writing the product fg as a quotient:
This converts the given limit into an indeterminate form of type or ∞/∞, so that we can use L’Hospital’s Rule.

16. INDETERMINATE PRODUCTS
Example 4
Evaluate
The given limit is indeterminate because, as x → 0+, the first factor (x) approaches 0, whereas the second factor (ln x) approaches -∞.
So, L’Hospital’s Rule gives:

17. INDETERMINATE PRODUCTS
Note
In solving the example, another possible option would have been to write:
This gives an indeterminate form of the type 0/0.
However, if we apply L’Hospital’s Rule, we get a more complicated expression than the one we started with.

18. INDETERMINATE FORM—TYPE ∞ - ∞
If and , then
the limit
is called an indeterminate form of type ∞ - ∞.

19. INDETERMINATE DIFFERENCES
Example 5
Compute
First, notice that sec x → ∞ and tan x → ∞ as x → (π/2)-.
So, the limit is indeterminate ∞ - ∞.

20. INDETERMINATE DIFFERENCES
Example 5
Here, we use a common denominator:
Note that the use of L’Hospital’s Rule is justified because 1 – sin x → 0 and cos x → 0 as x → (π/2)-.