# Introduction to Limits

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OBJECTIVES:
1. Use the definition of limit to estimate limits.
2. Determine whether limits of functions exist.
3. Use properties of limits and direct substitution to evaluate limits.
1. 12 Limits and an Introduction
to Calculus
2. 12.1 Introduction to Limits
3.  Use the definition of limit to estimate limits.
 Determine whether limits of functions exist.
 Use properties of limits and direct substitution to
evaluate limits.
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4. The Limit Concept
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5. The Limit Concept
The notion of a limit is a fundamental concept of calculus.
In this chapter, you will learn how to evaluate limits and
how to use them in the two basic problems of calculus: the
tangent line problem and the area problem.
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6. Example 1 – Finding a Rectangle of Maximum Area
Find the dimensions of a rectangle that has a perimeter of
24 inches and a maximum area.
Let w represent the width of the rectangle and let l
represent the length of the rectangle. Because
2w + 2l = 24 Perimeter is 24.
it follows that l = 12 – w,
as shown in the figure.
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7. Example 1 – Solution cont’d
So, the area of the rectangle is
A = lw Formula for area
= (12 – w)w Substitute 12 – w for l.
= 12w – w2. Simplify.
Using this model for area, experiment with different values
of w to see how to obtain the maximum area.
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8. Example 1 – Solution cont’d
After trying several values, it appears that the maximum
area occurs when w = 6, as shown in the table.
In limit terminology, you can say that “the limit of A as w
approaches 6 is 36.” This is written as
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9. Definition of Limit
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10. Definition of Limit
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11. Example 2 – Estimating a Limit Numerically
Use a table to estimate numerically the limit: .
Let f (x) = 3x – 2.
Then construct a table that shows values of f (x) for two
of x-values—one set that approaches 2 from the left and
one that approaches 2 from the right.
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12. Example 2 – Solution cont’d
From the table, it appears that the closer x gets to 2, the
closer f (x) gets to 4. So, you can estimate the limit to be 4.
Figure 12.1 illustrates this conclusion.
Figure 12.1
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13. Limits That Fail to Exist
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14. Limits That Fail to Exist
Next, you will examine some functions for which limits do
not exist.
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15. Example 6 – Comparing Left and Right Behavior
Show that the limit does not exist.
Consider the graph of f (x) = | x |/x.
From Figure 12.4, you can see that
for positive x-values
and for negative x-values
Figure 12.4
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16. Example 6 – Solution cont’d
This means that no matter how close x gets to 0, there will
be both positive and negative x-values that yield f (x) = 1
and f (x) = –1.
This implies that the limit does not exist.
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17. Limits That Fail to Exist
Following are the three most common types of behavior
associated with the nonexistence of a limit.
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18. Properties of Limits and
Direct Substitution
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19. Properties of Limits and Direct Substitution
You have seen that sometimes the limit of f (x) as x  c is
simply f (c), as shown in Example 2. In such cases, the limit
can be evaluated by direct substitution.
That is,
Substitute c for x.
There are many “well-behaved” functions, such as
polynomial functions and rational functions with nonzero
denominators, that have this property.
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20. Properties of Limits and Direct Substitution
The following list includes some basic limits.
This list can also include trigonometric functions. For
and
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21. Properties of Limits and Direct Substitution
By combining the basic limits with the following operations,
you can find limits for a wide variety of functions.
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22. Example 9 – Direct Substitution and Properties of Limits
Find each limit.
a. b. c.
d. e. f.
Use the properties of limits and direct substitution to
evaluate each limit.
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23. Example 9 – Solution cont’d
b. Property 1
c. Property 4
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24. Example 9 – Solution cont’d
e. Property 3
f. Properties 2 and 5
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25. Properties of Limits and Direct Substitution
Example 9 shows algebraic solutions. To verify the limit in
Example 9(a) numerically, for instance, create a table that
shows values of x2 for two sets of x-values—one set that
approaches 4 from the left and one that approaches 4 from
the right, as shown below.
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26. Properties of Limits and Direct Substitution
From the table, you can see that the limit as x approaches
4 is 16. To verify the limit graphically, sketch the graph
of y = x2. From the graph shown in Figure 12.7, you can
determine that the limit as x approaches 4 is 16.
Figure 12.7
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27. Properties of Limits and Direct Substitution
The following summarizes the results of using direct
substitution to evaluate limits of polynomial and rational
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