Introduction to Limits

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Sharp Tutor Tue, Jan 18, 2022 05:59 PM UTC

This presentation provides a brief introduction to limits.

1. Pre-Calculus Honors
11.1: Introduction to Limits
Copyright © Cengage Learning. All rights reserved.

2. Example 1 – Finding a Rectangle of Maximum Area
You are given 24 inches of wire and are asked to
form a rectangle whose area is as large as possible.
What dimensions should the rectangle have?
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3. Example 1 – Solution
Let w represent the width of the rectangle and let l
represent the length of the rectangle.
Because 2w + 2l = 24, it follows that l = 12 – w
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4. Example 1 – Solution cont’d
So, the area of the rectangle is
A = lw
A = (12 – w)w
A = 12w – w2.
Using this model for area, you can experiment with different
values of w to see how to obtain the maximum area.
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5. Example 1 – Solution cont’d
After trying several values, it appears that the
maximum area occurs when w = 6 as shown in the
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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lim A lim(12 w  w ) 36.
w 6 w 6
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6. Definition of Limit
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7. Example 2 – Estimating a Limit Numerically
1.) USING A TABLE TO ESTIMATE A LIMIT.
Use a table to estimate
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8. Example 2 – Estimating a Limit Numerically
1.) USING A TABLE TO ESTIMATE A LIMIT.
Let f (x) = 3x – 2.
Construct a table that shows values of f (x) for two
sets of x-values—one set that approaches 2 from the
left and one that approaches 2 from the right.
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. 8

9. Example 2 – Solution cont’d
The graph adds further support to this conclusion.
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10. Example 3 – Estimating a Limit Numerically
2.) USING A TABLE TO ESTIMATE A LIMIT.
x
Use a table to estimate lim
x 0 x 1  1
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11. Example 3 – Estimating a Limit Numerically
Reinforce with the graph.
x
lim
x 0 x 1  1
f(x) has a limit as x  0
even though the function
is not defined at x = 0.
The existence or nonexistence of f(x) at x = c has no
bearing on the existence of the limit of f(x) as x
approaches c.
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12. Examples 4 and 5 – Estimating a Limit Graphically
3 2
x  x x 1
3.) Estimate the limit graphically: lim
x 1 x 1
4.) Use the graph to find the limit of f(x) as x
approaches 3, where f is defined as:
2, x 3
f ( x) 
0, x 3
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13. Pre-Calculus Honors
11.1: Introduction to Limits
HW: p.757-758 (6, 8, 18, 26, 30, 34-40 even)
Copyright © Cengage Learning. All rights reserved.
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14. Limits That Fail to Exist
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15. Example 6 – Comparing Left and Right Behavior
Show that the limit does not exist by analyzing the graph.
x
1.) lim
x 0 x
1
2.) lim 2
x 0 x
1
3.) lim sin
x 0 x
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17. Properties of Limits and Direct Substitution
You have seen that sometimes the limit of f (x) as x →
c is simply f (c). In such cases, it is said that 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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18. Properties of Limits and Direct Substitution
Some of the basic ones are included in the following list.
Also true for trig functions. 18

19. Properties of Limits and Direct Substitution
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20. Example 9 – Direct Substitution and Properties of Limits
a. lim x 2 Direct Substitution
x 4
b. lim 5 x Scalar Multiple Property
x 4
tan x
c. lim Quotient Property
x  x
d. lim x Direct Substitution
x 9
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21. Example 9 – Direct Substitution and Properties of Limits
e. lim( x cos x)
x 
Product Property
f. lim( x  4) 2
x 3
Sum and Power Properties
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22. Properties of Limits and Direct Substitution
The results of using direct substitution to
evaluate limits of polynomial and rational
functions are summarized as follows.
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