12.1.4 More Exotic Examples of Indeterminate Forms

Question 1

More Exotic Examples of Indeterminate Forms

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

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

Answer 3

Finding the derivative of (2x)

Question 4

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

Answer 4

limx→0 87x3+6sinx+10

Question 5

Evaluate limx→∞ 2x+5ex.

Answer 5

0

Question 6

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

Answer 6

The limit is equal to 1

Question 7

Evaluate limx→0cos(10x)−110x

Answer 7

0

Question 8

Evaluate limx→0sin2xcosx−1

Answer 8

−2

Question 9

Evaluate limx→0 1/sinx.

Answer 9

The limit does not exist.

Question 10

Evaluate limx→0 cos2x−1/sinx

Answer 10

0

Question 11

Evaluate limx→∞(lnx)+3/2−x3

Answer 11

0

Question 12

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

Answer 12

b

Question 13

Evaluate limx→−∞ x4/e−x

Answer 13

0

Question 14

Evaluate limx→π/2 sinx+cosx−1/cosx

Answer 14

1