Lesson 19 Limits Involving Infinity



Lesson 20 Infinite Limits

Example Find [pic]

Let [pic]

|x |f(x) | |x |f(x) |

|3.1 |50 | |2.9 |( 50 |

|3.01 |500 | |2.99 |( 500 |

|3.001 |5000 | |2.999 |( 5000 |

|3.0001 |50,000 | |2.9999 |( 50,000 |

Definition Let f be a function defined on the interval (a, a + r), r > 0. (Right-side of x = a) If as x ( a+, f(x) gets “larger and larger positively”, then we write [pic].

Definition Let f be a function defined on the interval (a, a + r), r > 0. (Right-side of x = a) If as x ( a+, f(x) gets “larger and larger negatively”, then we write [pic].

Definition Let f be a function defined on the interval (a ( r, a), r > 0. (Left-side of x = a)

If as x ( a(, f(x) gets “larger and larger positively”, then we write [pic].

Definition Let f be a function defined on the interval (a ( r, a), r > 0. (Left-side of x = a)

If as x ( a(, f(x) gets “larger and larger negatively”, then we write [pic].

Definition [pic] if and only if [pic] and [pic]

Definition [pic] if and only if [pic] and [pic]

Thus, for our example above, we may write [pic], [pic], and

[pic] Does Not Exist (DNE).

NOTE: If a one-sided limit of a function has the form [pic], then the answer to the limit will either be ( or ( (. The sign of the infinity will depend on the sign of the function.

Examples Find the following limits.

1. [pic]

[pic]

The answer to this one-sided limit is either ( or ( (. We need to find the sign of the function [pic] on the immediate right side of 3.

Sign of [pic]: X +

(

3

Answer: (

2. [pic]

[pic]

The answer to this one-sided limit is either ( or ( (. We need to find the sign of the function [pic] on the immediate left side of 3.

Sign of [pic]: ( X

(

3

Answer: ( (

3. [pic]

[pic]

The answer for this limit is either ( , ( ( , or DNE. You need to calculate the two one-sided limits. NOTE: An answer of ( still tells us the limit does not exist. However, it provides more information. Namely, that the two one-sided limits go off to the same signed infinity. This is also true for an answer of ( (.

From Example 1 above, we have that [pic]. From Example 2 above, we have that [pic]. Since the one-sided limits go off to different signed infinities, then we say that [pic] = DNE.

Answer: DNE

4. [pic]

[pic]

The answer for this limit is either ( , ( ( , or DNE. You need to calculate the two one-sided limits. We need to find the sign of the function [pic] on the immediate left and right sides of ( 8.

Sign of [pic]: X + +

( (

- 10 - 8

Thus, [pic] and [pic]. Since the signed infinities are the same, then we say that [pic]

Answer: (

5. [pic]

[pic]

The answer to this one-sided limit is either ( or ( (. We need to find the sign of the function [pic] on the immediate left side of 4.

Sign of [pic]: X X ( X X

( ( ( (

- 4 0 4 5

Thus, [pic]

Answer: ( (

6. [pic]

[pic]

The answer to this one-sided limit is either ( or ( (. We need to find the sign of the function [pic] on the immediate right side of 4.

Sign of [pic]: X X X + X

( ( ( (

- 4 0 4 5

Thus, [pic]

Answer: (

7. [pic]

[pic]

The answer for this limit is either ( , ( ( , or DNE. You need to calculate the two one-sided limits. We need to find the sign of the function [pic] on the immediate left and right sides of 3.

Sign of [pic]: ( ( X

( (

3. 9

Thus, [pic] and [pic]. Since the signed infinities are the same, then we say that [pic]

Answer: ( (

8. [pic]

[pic] = [pic]

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

The answer for this limit is either ( , ( ( , or DNE. You need to calculate the two one-sided limits. We need to find the sign of the function [pic] on the immediate left and right sides of [pic].

Sign of [pic]: + ( X

( (

[pic] 0

Thus, [pic] and [pic].

Thus, [pic] = DNE

Answer: DNE

Lesson 20 Limits At Infinity

Example Consider the function [pic] for “large” values of x.

|x |f(x) | |x |f(x) |

|10 |- 1.87 | |- 10 |- 2.07 |

|100 |- 1.9897 | |- 100 |- 2.0097 |

|1000 |-1.998997 | |- 1000 |- 2.000997 |

|10,000 |- 1.99989997 | |- 10,000 |- 2.00009997 |

|100,000 |- 1.9999899997 | |- 100,000 |- 2.0000099997 |

|1,000,000 |- 1.999998999997 | |- 1,000,000 |- 2.000000999997 |

Definition Let f be a function defined on the interval (a, [pic]), where a is a real number. The statement [pic] means as x approaches (positive) infinity (“as x gets larger and larger positively”), f(x) approaches L.

Definition Let f be a function defined on the interval (([pic], b), where b is a real number. The statement [pic] means as x approaches negative infinity (“as x gets larger and larger negatively”), f(x) approaches L.

Thus, for our example above, we may write [pic] and [pic].

NOTE: [pic] for all [pic].

Theorem If r is a positive rational number and c is any nonzero real number, then [pic] and [pic] provided that [pic] is defined in the later case.

Examples Find the following limits.

1. [pic]

The largest exponent on the variable x in the numerator and denominator is 1. To change the form of the fraction [pic] in order to get fractions of the form in the theorem above, multiply the numerator and denominator of this fraction by 1 over x raised to this largest exponent of 1. That is, multiply the numerator and denominator of the fraction by [pic].

Thus,

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

Answer: 3

2. [pic]

The largest exponent on the variable x in the numerator and denominator is 2. To change the form of the fraction [pic] in order to get fractions of the form in the theorem above, multiply the numerator and denominator of this fraction by 1 over x raised to this largest exponent of 2. That is, multiply the numerator and denominator of the fraction by [pic]. Thus,

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

Answer: [pic]

3. [pic]

The largest exponent on the variable t in the numerator and denominator is 3. In order to change the form of the fraction [pic], multiply the numerator and denominator of this fraction by [pic]. Thus,

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

[pic] Answer: [pic]

4. [pic]

The largest exponent on the variable w in the numerator and denominator is 2. In order to change the form of the fraction [pic], multiply the numerator and denominator of this fraction by [pic]. Thus,

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

Answer: 0

5. [pic]

The largest exponent on the variable x in the numerator and denominator is 3. In order to change the form of the fraction [pic], multiply the numerator and denominator of this fraction by [pic]. Thus,

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

Since you can only approach positive infinity ([pic]) from one side (the left side), then this limit is a one-sided limit and the answer to the limit is either [pic] or [pic]. We must determine the sign of [pic] on the interval containing positive infinity.

Sign of [pic]: X X X +

( ( (

[pic] [pic] 1

Answer: [pic]

NOTE: Limits approaching negative infinity ([pic]) are also one-sided limits since you can only approach negative infinity from the right side.

6. [pic]

By continuity of the root function, we can pass the limit sign inside the radical sign. Thus, we have that

[pic] = [pic]

The largest exponent on the variable t in the numerator and denominator is 4. In order to change the form of the fraction [pic], multiply the numerator and denominator of this fraction by [pic] under the radical. Thus,

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

[pic] = [pic]

Answer: [pic]

7. [pic]

The largest exponent on the variable x in the numerator is 1. The largest exponent on the variable x in the denominator is 2, which is under the square root sign. Thus, you must apply the square root to this exponent by applying the square root to [pic]. Since x is approaching positive infinity, then [pic]. Thus, the largest exponent on the variable x in the denominator is 1. In order to change the form of the fraction [pic], multiply the numerator and denominator of this fraction by [pic]. Remember, you will have to square the fraction [pic] in order to pass it under the square root sign. Thus,

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

[pic]

Answer: [pic]

8. [pic]

The largest exponent on the variable x in the numerator is 12, which is under the fourth root sign. Thus, you must apply the fourth root to this exponent by applying the fourth root to [pic]. Since x is approaching negative infinity, then [pic]. Thus, the largest exponent on the variable x in the numerator is 3. The largest exponent on the variable x in the denominator is 3. In order to change the form of the fraction [pic], multiply the numerator and denominator of this fraction by [pic]. Remember, you will have to raise the fraction [pic] to the fourth power in order to pass it under the fourth root sign. Thus,

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

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

Answer: [pic]

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