Lesson 20 Infinite Limits
Example Find
Let
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 xa+, f(x) gets “larger and larger positively”, then we write .
Definition Let f be a function defined on the interval (a, a + r), r > 0. (Right-side of x = a) If as xa+, f(x) gets “larger and larger negatively”, then we write .
Definition Let f be a function defined on the interval (ar, a), r > 0. (Left-side of x = a)
If as xa, f(x) gets “larger and larger positively”, then we write .
Definition Let f be a function defined on the interval (ar, a), r > 0. (Left-side of x = a)
If as xa, f(x) gets “larger and larger negatively”, then we write .
Definition if and only if and
Definition if and only if and
Thus, for our example above, we may write , , and
Does Not Exist (DNE).
NOTE: If a one-sided limit of a function has the form , 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.
The answer to this one-sided limit is either or . We need to find the sign of the function on the immediate right side of 3.
Sign of : X+
3
Answer:
2.
The answer to this one-sided limit is either or . We need to find the sign of the function on the immediate left side of 3.
Sign of : X
3
Answer:
3.
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 . From Example 2 above, we have that . Since the one-sided limits go off to different signed infinities, then we say that = DNE.
Answer: DNE
4.
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 on the immediate left and right sides of 8.
Sign of : X++
- 10 - 8
Thus, and . Since the signed infinities are the same, then we say that
Answer:
5.
The answer to this one-sided limit is either or . We need to find the sign of the function on the immediate left side of 4.
Sign of : XXXX
- 4 0 45
Thus,
Answer:
6.
The answer to this one-sided limit is either or . We need to find the sign of the function on the immediate right side of 4.
Sign of : XXX+X
- 4 0 45
Thus,
Answer:
7.
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 on the immediate left and right sides of 3.
Sign of : X
39
Thus, and . Since the signed infinities are the same, then we say that
Answer:
8.
=
= = =
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 on the immediate left and right sides of .
Sign of : + X
0
Thus, and .
Thus, = DNE
Answer: DNE
Lesson 20 Limits At Infinity
Example Consider the function 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, ), where a is a real number. The statement 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 (, b), where b is a real number. The statement 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 and .
NOTE: for all .
Theorem If r is a positive rational number and c is any nonzero real number, then and provided that is defined in the later case.
Examples Find the following limits.
1.
The largest exponent on the variable x in the numerator and denominator is 1. To change the form of the fraction 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 .
Thus,
= = =
Answer: 3
2.
The largest exponent on the variable x in the numerator and denominator is 2. To change the form of the fraction 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 . Thus,
= = =
Answer:
3.
The largest exponent on the variable t in the numerator and denominator is 3. In order to change the form of the fraction , multiply the numerator and denominator of this fraction by . Thus,
= = = =
Answer:
4.
The largest exponent on the variable w in the numerator and denominator is 2. In order to change the form of the fraction , multiply the numerator and denominator of this fraction by . Thus,
= = =
Answer: 0
5.
The largest exponent on the variable x in the numerator and denominator is 3. In order to change the form of the fraction , multiply the numerator and denominator of this fraction by . Thus,
= = =
Since you can only approach positive infinity () from one side (the left side), then this limit is a one-sided limit and the answer to the limit is either or . We must determine the sign of on the interval containing positive infinity.
Sign of : XXX+
1
Answer:
NOTE: Limits approaching negative infinity () are also one-sided limits since you can only approach negative infinity from the right side.
6.
By continuity of the root function, we can pass the limit sign inside the radical sign. Thus, we have that
=
The largest exponent on the variable t in the numerator and denominator is 4. In order to change the form of the fraction , multiply the numerator and denominator of this fraction by under the radical. Thus,
= = = =
=
Answer:
7.
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 . Since x is approaching positive infinity, then . Thus, the largest exponent on the variable x in the denominator is 1. In order to change the form of the fraction , multiply the numerator and denominator of this fraction by . Remember, you will have to square the fraction in order to pass it under the square root sign. Thus,
= = = =
Answer:
8.
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 . Since x is approaching negative infinity, then . 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 , multiply the numerator and denominator of this fraction by . Remember, you will have to raise the fraction to the fourth power in order to pass it under the fourth root sign. Thus,
= = =
= = =
Answer:
Copyrighted by James D. Anderson, The University of Toledo