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Chapter 2 | Limits
2.3 | The Limit Laws
Learning Objectives
2.3.1 Recognize the basic limit laws. 2.3.2 Use the limit laws to evaluate the limit of a function. 2.3.3 Evaluate the limit of a function by factoring.
2.3.4 Use the limit laws to evaluate the limit of a polynomial or rational function. 2.3.5 Evaluate the limit of a function by factoring or by using conjugates. 2.3.6 Evaluate the limit of a function by using the squeeze theorem.
In the previous section, we evaluated limits by looking at graphs or by constructing a table of values. In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits. Evaluating Limits with the Limit Laws The first two limit laws were stated in Two Important Limits and we repeat them here. These basic results, together with the other limit laws, allow us to evaluate limits of many algebraic functions.
Theorem 2.4: Basic Limit Results For any real number a and any constant c ,
lim x lim x
x = a
(2.14) (2.15)
i.
→ a
c = c
ii.
→ a
Example 2.13 Evaluating a Basic Limit
Evaluate each of the following limits using Basic Limit Results . a. lim x →2 x b. lim x →2 5
Solution a. The limit of x as x approaches a is a : lim x →2
x =2.
b. The limit of a constant is that constant: lim x →2
5=5.
We now take a look at the limit laws , the individual properties of limits. The proofs that these laws hold are omitted here.
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