section 2.3
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Section 2.3. Evaluating Limits Analytically. Theorems Involving Limits. Theorems Involving Limits. Theorems Involving Limits (cont.). Theorems Involving Limits. Example 1. Find. Example 2. Find. Example 3. Find. Example 4. Find. Other Theorems Involving Limits. - PowerPoint PPT PresentationTRANSCRIPT
SECTION 2.3Evaluating Limits Analytically
Theorems Involving Limits
Theorem 2.1 Some Basic Limits (p. 79)
Let and be real numbers and let be a positive integer.
1. 2. 3.
Theorems Involving Limits
Theorem 2.2 Properties of Limits (p. 79)
Let and be real numbers, let be a positive integer, and let and be functions.
1. Scalar Multiple:
2. Sum or difference:
3. Product:
4. Quotient: , provided
5. Power:
Theorems Involving Limits (cont.)
Theorem 2.3 Limits of Polynomial and Rational Functions (p. 80)
If is a polynomial and is a real number, then
.
If is a rational function given by and is a real number such that , then
.
Theorems Involving LimitsTheorem 2.4 The Limit of a Function Involving a Radical(p. 80)
Let be a positive integer. The following limit is valid for all if is odd, and is valid for if is even.
Theorem 2.5 The Limit of a Composite Function (p. 81)
Example 1Find .
Example 2Find .
Example 3Find .
Example 4Find .
Other Theorems Involving Limits
• Theorem 2.6 deals with finding the limits of trigonometric, exponential, and logarithmic functions.
• Theorem 2.7 talks about fnc.’s that agree at all but one point.
• Theorem 2.8 is the Squeeze Theorem.
Example 5Find .
Theorem 2.9
1. 2. 3. Find
Example 5
Find given .
Limits of Transcendental Functions
Example 6Find the limit if it exists.
a.
b.
Functions Agreeing at All But One Point
Example 7Find .