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Mathematics /AP Calculus AB: 12.1.4 More Exotic Examples of Indeterminate Forms

AP Calculus AB: 12.1.4 More Exotic Examples of Indeterminate Forms

Mathematics14 CardsCreated 7 months ago

This content explores complex uses of L’Hôpital’s Rule, especially for limits involving products, compositions, and trigonometric functions. It highlights when and how to apply the rule properly, warns against using it without an actual indeterminate form, and reinforces the need for product and chain rule in derivative calculations.

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More Exotic Examples of Indeterminate Forms

As long as the limit still produces an indeterminate form, you can reuse L’Hôpital’s rule.
When applying L’Hôpital’s rule to a quotient containing one or more products or compositions of functions, it is necessary to use the product or chain rules.
L’Hôpital’s rule might not give you the right answer if you use it on a limit that does not produce an indeterminate form.

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Definition

More Exotic Examples of Indeterminate Forms

As long as the limit still produces an indeterminate form, you can reuse L’Hôpital’s rule.
When applying L’Hôpital’s rule to...

note

Some limits produce an indeterminate form that cannot be eliminated by factoring. In these cases, L’Hôpital’s rule is very useful.

Which of the following is not a step when L’Hôpital’s rule is used to determine limx→2 x2−x−2/x−2?

Finding the derivative of (2x)

Which of the following limits does not produce an indeterminate form?

limx→0 87x3+6sinx+10

Evaluate limx→∞ 2x+5ex.

0

Which of the following statements about this limit expression is not correct?
limx→0 tanx/2x

The limit is equal to 1

Evaluate limx→0cos(10x)−110x

0

Evaluate limx→0sin2xcosx−1

-2

Evaluate limx→0 1/sinx.

The limit does not exist.

Evaluate limx→0 cos2x−1/sinx

0

Evaluate limx→∞(lnx)+3/2−x3

0

How many times is L’Hôpital’s rule used to solve for limx→∞ xb/ex, where b is a positive integer?

b

Evaluate limx→−∞ x4/e−x

0

Evaluate limx→π/2 sinx+cosx−1/cosx

1

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AP Calculus AB: 12.1.4 More Exotic Examples of Indeterminate Forms

Term	Definition
More Exotic Examples of Indeterminate Forms	As long as the limit still produces an indeterminate form, you can reuse L’Hôpital’s rule. When applying L’Hôpital’s rule to a quotient containing one or more products or compositions of functions, it is necessary to use the product or chain rules. L’Hôpital’s rule might not give you the right answer if you use it on a limit that does not produce an indeterminate form.
note	Some limits produce an indeterminate form that cannot be eliminated by factoring. In these cases, L’Hôpital’s rule is very useful. These two limits are classic limits that may appear in other situations, such as the limit definition of the derivative for trig functions. To apply L’Hôpital’s rule, you will need to remember the derivatives of sin x and cos x. This limit does not meet the criteria for L’Hôpital’s rule because it does not produce an indeterminate form. If you tried to use L’Hôpital’s rule here, you would get a different answer. In a complicated limit it can be helpful to think about the behavior of specific terms. In this example, the 3 has a negligible effect. The x-squared term in the numerator will overpower x ln x in the denominator. After using L’Hôpital’s rule once, the limit produces an indeterminate form again. A second application results in an answer. You can say the limit is infinity, but since that is not a number you can also say that it does not exist.
Which of the following is not a step when L’Hôpital’s rule is used to determine limx→2 x2−x−2/x−2?	Finding the derivative of (2x)
Which of the following limits does not produce an indeterminate form?	limx→0 87x3+6sinx+10
Evaluate limx→∞ 2x+5ex.	0
Which of the following statements about this limit expression is not correct? limx→0 tanx/2x	The limit is equal to 1
Evaluate limx→0cos(10x)−110x	0
Evaluate limx→0sin2xcosx−1	-2
Evaluate limx→0 1/sinx.	The limit does not exist.
Evaluate limx→0 cos2x−1/sinx	0
Evaluate limx→∞(lnx)+3/2−x3	0
How many times is L’Hôpital’s rule used to solve for limx→∞ xb/ex, where b is a positive integer?	b
Evaluate limx→−∞ x4/e−x	0
Evaluate limx→π/2 sinx+cosx−1/cosx	1