Download presentation
Presentation is loading. Please wait.
1
We will find limits algebraically
1.2 Computing Limits We will find limits algebraically
2
Theorem 1.2.1 Let a and k be real numbers.
3
Theorem 1.2.1 Let a and k be real numbers.
4
Theorem 1.2.1 Let a and k be real numbers.
5
Theorem 1.2.1 Let a and k be real numbers.
6
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
7
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
8
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then: The limit of a sum is the sum of the limits.
9
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
10
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then: The limit of a difference is the difference of the limits.
11
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
12
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then: The limit of a product is the product of the limits
13
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
14
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then: The limit of a quotient is the quotient of the limits
15
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
16
Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then: The limit of an nth root is the nth root of the limit.
17
Constants A constant factor can be moved through a limit symbol.
18
Example 5 Find
19
Example 5 Find
20
Example 5 Find
21
Theorem 1.2.3 For any polynomial and any real number a,
22
Rational Functions We will have three situations when dealing with rational functions. The denominator is not zero.
23
Rational Functions We will have three situations when dealing with rational functions. The denominator is not zero. The denominator and the numerator are both zero.
24
Rational Functions We will have three situations when dealing with rational functions. The denominator is not zero. The denominator and the numerator are both zero. The denominator is zero and the numerator is not.
25
Denominator not Zero When the denominator is not zero, we evaluate the numerator and denominator and for this problem use theorem 1.2.2d: the limit of a quotient is the quotient of the limit.
26
Denominator not Zero When the denominator is not zero, we evaluate the numerator and denominator and for this problem use theorem 1.2.2d: the limit of a quotient is the quotient of the limit.
27
Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph. Even though the value doesn’t exist, the limit does.
28
Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph. Factor, cancel, evaluate.
29
Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph. Factor, cancel, evaluate.
30
Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph. Factor, cancel, evaluate.
31
Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph. Factor, cancel, evaluate.
32
Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph. Rationalize the denominator, cancel, evaluate.
33
Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote.
34
Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote. Sign analysis- Determine what the graph is doing around the asymptote.
35
Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote. Sign analysis ___+___|_____-_____|___+___|___-___
36
Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote. Sign analysis ___+___|_____-_____|___+___|___-___
37
Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote. Sign analysis Remember, all three answers mean that the limit does not exist. The first two answers tell us exactly what the function is doing and why the limit DNE.
38
Limits of Piecewise Functions
Let Find
39
Limits of Piecewise Functions
Let Find
40
Limits of Piecewise Functions
Let Find
41
Limits of Piecewise Functions
Let Find
42
Homework Page 87 1-31 odd
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.