LESSON X - Mathematics & Statistics



LESSON 11 L’HOPITAL’S RULE AND THE SANDWICH THEOREM

Theorem (Cauchy’s Formula) If the functions f and g are continuous on the closed interval [pic] and differentiable on the open interval [pic] and if [pic] for all x in [pic], then there is a number c in [pic] such that

[pic].

Proof Note that [pic]. For if [pic], then [pic]. Thus, by Rolle’s Theorem, there exists a number w in the open interval [pic] such that [pic]. This would contradict the fact that [pic] for all x in [pic].

Define the function h on the closed interval [pic] by

[pic].

Note that [pic] and [pic] are fixed numbers. Since the functions f and g are continuous on the closed interval [pic], then the function h is continuous on [pic]. Since the functions f and g are differentiable on the open interval [pic], then the function h is differentiable on [pic].

Note that [pic] =

[pic] = [pic] and

[pic] =

[pic] = [pic]

Thus, by Rolle’s Theorem, there exists a number c in the open interval [pic] such that [pic]. Since [pic] and [pic] and [pic] are constants, then [pic].

Thus, [pic] and [pic]

[pic]

[pic] [pic]

NOTE: Cauchy’s Formula is a generalization of the Mean Value Theorem. If [pic], then [pic], [pic], and [pic] for all x. Thus, [pic] [pic] [pic].

Theorem (L’Hopital’s Rule) Suppose the functions f and g are differentiable on a deleted neighborhood of the number a. If [pic] for all [pic] and if [pic] has the indeterminate form [pic] or [pic] , then

[pic].

provided that [pic] exists or [pic].

Proof Let [pic] = [pic] be a deleted neighborhood of the number a on which the functions f and g are differentiable. Suppose that [pic] has the indeterminate form [pic] at [pic]. Then we have that [pic] and [pic]. Suppose that [pic] for some number L. We want to show that [pic]. Let’s defined the following two functions F and G on the open interval [pic].

[pic] and [pic]

Since [pic] for all [pic] in [pic], then the function F is differentiable on the deleted neighborhood [pic]. Thus, the function F is continuous on the deleted neighborhood [pic]. Since [pic] = [pic], then the function F is continuous at [pic]. Thus, the function F is continuous on the interval [pic]. Similarly, the function G is continuous on [pic]. Thus, for each x in the deleted neighborhood [pic], we can apply Cauchy’s Formula to the interval [pic] if [pic] or to the interval [pic] if [pic] for the functions F and G. Thus, by Cauchy’s Formula, there exists a number [pic] between a and x such that

[pic]

Since [pic], then we have that [pic] and [pic]. Since [pic], then we have that [pic] and [pic]. Also, [pic] and [pic]. Thus,

[pic] [pic] [pic].

Since [pic] is between a and x, then as [pic], we have that [pic]. Thus,

[pic].

This argument would also be true if [pic]. The argument for the indeterminate form [pic] is more difficult and can be found in texts on advanced calculus.

COMMENT: One common mistake, which is made by students applying L’Hopital’s Rule, is using the Quotient Rule instead of differentiating the numerator and denominator. The other mistake is applying L’Hopital’s Rule without verifying that you have the indeterminate form [pic] or [pic].

Examples Find the following limits, if they exist.

1. [pic]

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = 1

Answer: 1

NOTE: The calculation of this limit was needed in order to calculate the derivative of the sine function.

2. [pic]

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic]

Answer: [pic]

3. [pic]

Since [pic], then [pic] = [pic].

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] =

[pic] = [pic] = [pic] = 0

Answer: 0

4. [pic]

[pic] = [pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic]

Answer: 0

NOTE: The calculation of this limit was also needed in order to calculate the derivative of the sine function.

5. [pic]

[pic] = [pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic]

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule again, we have that

[pic] = [pic] = [pic] = [pic]

Answer: [pic]

6. [pic]

[pic] = [pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic]

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule again, we have that

[pic] = [pic] = [pic]

[pic] = [pic] This is NOT an indeterminate form.

Sign of [pic]: [pic]

( (

[pic] 0

Thus, [pic]. Since [pic] = [pic] by L’Hopital’s Rule, then [pic].

Thus, [pic] = [pic].

Answer: [pic]

7. [pic]

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic] 4

Answer: 4

8. [pic]

Since the tangent function is an odd function, then we may write [pic] =

[pic]. Thus, we have that

[pic] = [pic] = [pic]

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic]

[pic] = [pic] This is NOT an indeterminate form.

Sign of [pic]: + +

(

0

Thus, [pic] and [pic]. Thus, [pic].

Since [pic] = [pic] by L’Hopital’s Rule, then [pic].

Thus, [pic] = [pic].

Answer: [pic]

9. [pic]

[pic] = [pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that [pic] =

[pic]

Now, using Properties of Limits, we have that [pic] =

[pic] = [pic]

Since [pic], then [pic] =

[pic] = [pic]

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule again, we have that

[pic] = [pic] = [pic] = [pic] = [pic]

Thus, [pic] = [pic] = [pic] = [pic].

Answer: [pic]

10. [pic]

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule again, we have that

[pic] = [pic] = [pic]

Answer: [pic]

11. [pic]

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic]

Answer: 0

12. [pic]

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic]

Answer: [pic]

13. [pic]

[pic] = [pic] Indeterminate Form

You can apply L’Hopital’s Rule to this limit. However, it much easier to use the methods of the MATH-1850 course to evaluate this limit.

[pic] = [pic] =

[pic] = [pic] =

[pic] = [pic] =

[pic] = [pic] = [pic]

Answer: [pic]

NOTE: I will let you do the TEN applications of L’Hopital’s Rule in order to produce this answer.

14. [pic]

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic] =

[pic]

Answer: [pic]

NOTE: To calculate this limit, without using L’Hopital’s Rule, would require the algebra of rationalizing both the numerator and the denominator.

15. [pic]

[pic] = [pic] = [pic] =

[pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] =

[pic] = [pic] = [pic] = [pic] = [pic] =

[pic]

Answer: [pic]

NOTE: This is the limit which is required to calculate the value of the derivative of the function [pic] at [pic] using the definition of derivative. It is ironic that we used the value of the derivative of this function at [pic] in order to calculate this limit. To calculate this limit, without using L’Hopital’s Rule, would require the algebra of rationalizing the numerator. Since [pic], then in order to rationalize the numerator of [pic] in the fraction [pic], we would need to multiply the numerator and denominator of the fraction by [pic] =

[pic]. Thus, we would have that

[pic] =

[pic] =

[pic] =

[pic] =

[pic] =

[pic] =

[pic] =

[pic] = [pic] =

[pic] = [pic] =

[pic] = [pic] = [pic] = [pic] = [pic] =

[pic] L’Hopital’s Rule was definitely easier.

16. [pic]

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] =

[pic] = [pic] = [pic] =

[pic] = [pic] = [pic] = [pic]

Answer: [pic]

17. [pic]

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] =

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule again, we have that

[pic] = [pic] =

[pic] = [pic] = [pic] = [pic]

Answer: [pic]

18. [pic]

[pic] = [pic] Indeterminate Form

Since [pic] = [pic], then applying L’Hopital’s Rule to the last limit, we have that

[pic] = [pic] = [pic]

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule again, we have that

[pic] = [pic] = [pic] =

[pic]

NOTE: By the double angle formula for sine, which is [pic] =

[pic], then [pic].

Answer: 0

19. [pic]

[pic] = [pic] Indeterminate Form

Since [pic] = [pic], then applying L’Hopital’s Rule to the last limit, we have that

[pic] = [pic] = [pic] =

[pic]

NOTE: By the double angle formula for sine, [pic].

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule again, we have that

[pic] = [pic] = [pic] = [pic] = [pic] = [pic]

Thus, [pic] = [pic] = [pic] =

[pic] = [pic]

Answer: [pic]

20. [pic]

[pic] = [pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic]

Since [pic] and [pic] = [pic] = [pic] = [pic] =

[pic], then [pic].

Thus, by L’Hopital’s Rule, we have that [pic].

Answer: [pic]

21. [pic]

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic] = [pic] = [pic]

Answer: [pic]

22. [pic]

[pic] = [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic]

Sign of [pic]: + [pic]

( (

3 [pic]

NOTE: The domain of the function [pic] is the set of numbers given by [pic].

Thus, [pic] and [pic].

Thus, [pic] = DNE.

Answer: DNE

Now, we will study the indeterminate forms of [pic], [pic], [pic], [pic], and [pic].

Using algebra, we may write the indeterminate form [pic] as either the indeterminate form [pic] or the indeterminate form [pic].

Using logarithmic functions, we may write the indeterminate forms [pic], [pic], and [pic] as the indeterminate form [pic].

Examples Find the following limits, if they exist.

1. [pic]

Since [pic], then [pic] = [pic] Indeterminate Form

Since [pic] = [pic], then we could write [pic] = [pic]. The indeterminate form of this last limit is [pic].

Since [pic] = [pic], then we could write [pic] = [pic]. The indeterminate form of this last limit is [pic].

Applying L’Hopital’s Rule to [pic], we have that

[pic] = [pic] = [pic]

Since [pic] = [pic], then we can apply L’Hopital’s Rule again. However, since the exponent on the x in the denominator will always stay negative and the numerator will always contain the exponential function, we will not be able to determine an answer for the limit.

Let’s apply L’Hopital’s Rule to [pic], which has the indeterminate form [pic]. Thus, we have that

[pic] = [pic] = [pic]

[pic] = [pic] Indeterminate Form

Apply L’Hopital’s Rule again, we have that

[pic] = [pic] = [pic] = [pic] =

[pic]

Thus, [pic] = [pic] = [pic] = [pic]

Answer: 0

2. [pic]

Since [pic] and [pic] = [pic] = 0, then [pic] = [pic] Indeterminate Form

Since [pic] = [pic] = [pic], then we can write

[pic] = [pic], which has an indeterminate form of [pic].

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] =

[pic] = [pic] = [pic] = [pic]

Answer: [pic]

3. [pic]

Since [pic] and [pic], then [pic] = [pic] Indeterminate Form

Since [pic] = [pic] = [pic], then we can write

[pic] = [pic], which has an indeterminate form of [pic].

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] =

[pic] = [pic]

[pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule again, we have that

[pic] = [pic]

Since [pic], then [pic] = [pic]

Answer: 0

4. [pic]

Since [pic], then [pic] = [pic] Indeterminate Form

Since [pic] = [pic] = [pic], then we can write [pic] = [pic], which has an indeterminate form of [pic].

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic]

Answer: 1

5. [pic]

Since [pic] = [pic] = [pic], then [pic] = [pic] Indeterminate Form

Since [pic] = [pic], then we can write [pic] = [pic], which has an indeterminate form of [pic].

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] =

[pic] = [pic] = [pic]

Answer: 6

6. [pic]

Since [pic] and [pic], then [pic] = [pic] = [pic] Indeterminate Form

[pic] = [pic] =

[pic]

Since [pic] and [pic], then [pic] = [pic] Indeterminate Form

Applying L’Hopital’s Rule to [pic], we have that

[pic] = [pic] = [pic]

Answer: 0

7. [pic]

Since [pic] and [pic], then [pic] = [pic] Indeterminate Form

[pic] = [pic] = [pic] =

[pic] = [pic]

NOTE: Since [pic] has an indeterminate form of [pic], then we can apply L’Hopital’s Rule.

Answer: [pic]

8. [pic]

Since [pic] and [pic], then [pic] = [pic] Indeterminate Form

[pic] =

[pic] =

[pic] = [pic] =

[pic] =

[pic] =

[pic] = [pic] = [pic] =

[pic] = [pic]

Answer: [pic]

9. [pic]

Since [pic] and [pic], then [pic] = [pic] Indeterminate Form

[pic] = [pic]

Now, consider [pic] = [pic] Indeterminate Form

Since [pic] = [pic], then applying L’Hopital’s Rule to this last limit, we have that

[pic] = [pic] = [pic]

Thus, [pic] = [pic].

Thus, [pic] = [pic].

Since [pic] and [pic], then [pic].

Thus, [pic] = [pic]

Answer: [pic]

10. [pic]

Since [pic] and [pic], then [pic] = [pic] Indeterminate Form

[pic] = [pic]

Now, consider [pic] = [pic] Indeterminate Form

Since [pic] = [pic], then applying L’Hopital’s Rule to this last limit, we have that

[pic] = [pic] = [pic] = [pic]

Thus, [pic]. Thus, [pic] = [pic].

Since [pic] and [pic] = 1, then [pic].

Thus, [pic] = [pic]

Answer: [pic]

Guidelines for evaluating [pic] if the indeterminate form is [pic], [pic], and [pic].

1. Let [pic]

2. Take the natural logarithm of both sides of the equation: [pic]

Of course, you can use any base for the logarithm. The natural logarithm is the easiest to use.

3. Find [pic] if it exists.

4. If [pic], then [pic] = [pic] = [pic].

NOTE: Since the natural exponential function [pic] is continuous for all real numbers and [pic] for all positive real numbers, then [pic] =

[pic] = [pic] = [pic] = [pic] = [pic].

Examples Find the following limits, if they exist.

1. [pic]

Since [pic] and [pic], then [pic] = [pic] Indeterminate Form.

Let [pic]. Then [pic] = [pic].

Now, find [pic], which has an indeterminate form of [pic].

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic]

Thus, [pic] = [pic]

Thus, [pic] = [pic] = [pic]

Answer: [pic]

2. [pic]

Since [pic] and [pic], then [pic] = [pic] Indeterminate Form.

Let [pic]. Then [pic] = [pic].

Now, find [pic], which has an indeterminate form of [pic].

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] =

[pic] = [pic]

Thus, [pic] = [pic] = [pic]

Thus, [pic] = [pic] = [pic]

Answer: [pic]

3. [pic]

[pic] = [pic] Indeterminate Form.

Let [pic]. Then [pic] = [pic].

Now, find [pic], which has an indeterminate form of [pic].

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] =

[pic] = [pic]

Thus, [pic] = [pic]

Thus, [pic] = [pic] = [pic]

Answer: [pic]

NOTE: Since [pic] = [pic], then we can use the expression [pic] to generate approximations for the number [pic]. You can use your calculator to verify that [pic].

4. [pic]

[pic] = [pic] Indeterminate Form.

Let [pic]. Then [pic] = [pic].

Now, find [pic], which has an indeterminate form of [pic].

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] =

[pic] = [pic] = [pic] = [pic]

Thus, [pic] = [pic]

Thus, [pic] = [pic] = [pic]

Answer: [pic] or [pic]

5. [pic]

[pic] = [pic] Indeterminate Form.

Let [pic]. Then [pic] = [pic].

Now, find [pic], which has an indeterminate form of [pic].

Applying L’Hopital’s Rule, we have that

[pic] = [pic] = [pic] = [pic]

Thus, [pic] = [pic]

Thus, [pic] = [pic] = [pic]

Answer: 1

6. [pic]

Since [pic] and [pic] = [pic], then [pic] = [pic] Indeterminate Form.

Let [pic]. Then [pic] = [pic].

Now, find [pic], which has an indeterminate form of [pic]. Since [pic] = [pic], which has an indeterminate form of [pic], then we can applying L’Hopital’s Rule to this last limit. Thus,

[pic] = [pic] = [pic] =

[pic] = [pic] = [pic]

Thus, [pic] = [pic]

Thus, [pic] = [pic] = [pic]

Answer: [pic]

7. [pic], where a is a constant

Since [pic] and [pic] = [pic], then [pic] = [pic] Indeterminate Form.

Let [pic]. Then [pic] = [pic].

Now, find [pic], which has an indeterminate form of [pic]. Since [pic] = [pic], which has an indeterminate form of [pic], then we can applying L’Hopital’s Rule to this last limit. Thus,

[pic] = [pic] = [pic] =

[pic] = [pic] = [pic]

Thus, [pic] = [pic]

Thus, [pic] = [pic] = [pic]

Answer: [pic]

Theorem (The Sandwich Theorem) If [pic] and [pic] and if [pic] for all x in a deleted neighborhood of [pic], then [pic].

Proof Let [pic]. In order to show that [pic], we need to show that there exists a [pic] such that [pic] whenever [pic].

Let [pic] = [pic] be a deleted neighborhood of [pic] for which [pic]. Thus, [pic] whenever [pic]

Since [pic], then there exists a [pic] such that [pic] whenever [pic]. Since [pic] [pic] [pic], then [pic] whenever [pic]. Thus, [pic] whenever [pic].

Since [pic], then there exists a [pic] such that [pic] whenever [pic]. Since [pic] [pic] [pic], then [pic] whenever [pic]. Thus, [pic] whenever [pic].

Let [pic] = min[pic]. Then [pic], [pic], and [pic].

Since [pic], then [pic]. Since [pic] whenever [pic], then [pic] whenever [pic].

Since [pic], then [pic]. Since [pic] whenever [pic], then [pic] whenever [pic].

Since [pic], then [pic]. Since [pic] whenever [pic], then [pic] whenever [pic].

Thus, [pic] whenever [pic]. Thus, [pic] whenever [pic]. Since [pic]

[pic] [pic]. Thus, [pic] whenever [pic]. Thus, [pic].

COMMENT: Some people use the phrase Squeeze Theorem instead of Sandwich Theorem.

Examples Find the following limits, if they exist.

1. [pic]

Since [pic] for all x and [pic] (since x is approaching positive infinity), then [pic] for all [pic].

Since [pic] and [pic], then [pic] by the Sandwich Theorem.

Answer: 0

2. [pic]

[pic] = [pic]

Since [pic] for all t and [pic] for t approaching negative infinity, then [pic] for all [pic].

Since [pic] and [pic], then [pic] by the Sandwich Theorem.

Thus, [pic] = [pic] = [pic]

Answer: 0

3. [pic]

[pic] = [pic]

NOTE: [pic] is defined for all [pic] and is undefined if [pic].

Since [pic] for all [pic] and [pic] (since x is approaching negative infinity), then [pic] [pic] for all [pic].

Since [pic] and [pic], then [pic] by the Sandwich Theorem.

Thus, [pic] = [pic] = [pic]

Answer: 0

4. [pic]

NOTE: [pic] and hence [pic] is defined for all [pic].

Since [pic] for all [pic], then [pic]. Since [pic] for [pic] approaching positive infinity, then [pic] for all [pic].

Since [pic] and [pic], then [pic] by the Sandwich Theorem.

Answer: 0

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