The Limit of a Function

1
Lesson 2-2

• The Limit of a Function

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Transparency 1-1
5-Minute Check on Algebra

1. 6x 45 18 3x
2. x2 45 4
3. (3x 4) (4x 7) 11
4. (4x 10) (6x 30) 180
5. Find the slope of the line k.
6. Find the
   slope of a perpendicular line to k

Standardized Test Practice
A
C
B
D
1/2
2
-1/2
-2
Click the mouse button or press the Space Bar to
display the answers.
3
Transparency 1-1
5-Minute Check on Algebra

1. 6x 45 18 3x
2. x2 45 4
3. (3x 4) (4x 7) 11
4. (4x 10) (6x 30) 180
5. Find the slope of the line k.
6. Find the
   slope of a perpendicular line to k

9x 45 18 9x -27 x -3
x² 49 x v49 x /- 7
7x - 3 11 7x 14 x 2
10x 20 180 10x 160 x 16
?y y2 y1 4 1
3 1 m ----- -----------
-------- ------ ---- ?x
x2 x1 6 0 6 2
?y
?x
Standardized Test Practice
A
C
B
D
1/2
2
-1/2
-2
Click the mouse button or press the Space Bar to
display the answers.
4
Objectives

• Determine and Understand one-sided limits
• Determine and Understand two-sided limits

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Vocabulary

• Limit (two sided) as x approaches a value a,
  f(x) approaches a value L
• Left-hand (side) Limit as x approaches a value
  a from the negative side, f(x) approaches a value
  L
• Right-hand (side) Limit as x approaches a value
  a from the positive side, f(x) approaches a value
  L
• DNE does not exist (either a limit
  increase/decreases without bound or the two
  one-sided limits are not equal)
• Infinity increases (8) without bound or
  decreases (-8) without bound NOT a number!!
• Vertical Asymptote at x a because a limit as
  x approaches a either increases or decreases
  without bound

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Homework Problem 1
t V m secant
0 1000 -50
5 694 -44.4 Estimates using Estimates using Estimates using
10 444 -38.8 1020 525 030
15 250 -33.3 -33.3 -33.3333
20 111 -27.8
25 28 -22.2
30 0 -16.6667
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Limits
When we look at the limit below, we examine the
f(x) values as x gets very close to a read
the limit of f(x), as x approaches a, equals
L One-Sided Limits Left-hand limit (as x
approaches a from the left side
smaller) RIght-hand limit (as x approaches a
from the right side larger) The two-sided
limit (first one shown) L if and only if both
one-sided limits L
if and only if and
lim f(x) L x?a
lim f(x) L x?a-
lim f(x) L x?a
lim f(x) L x?a
lim f(x) L x?a-
lim f(x) L x?a
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Vertical Asymptotes

• The line x a is called a vertical asymptote of
  y f(x) if at least one of the following is
  true

lim f(x) 8 x?a
lim f(x) 8 x?a-
lim f(x) 8 x?a
lim f(x) -8 x?a
lim f(x) -8 x?a-
lim f(x) -8 x?a
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Limits Using Graphs
One Sided Limits Limit from right lim f(x)
5 x?10 Limit from left lim
f(x) 3 x?10- Since the two
one-sided limits are not equal, then lim f(x)
DNE x?10
Usually a reasonableguess would be lim
f(x) f(a) x?a (this will be
true forcontinuous functions) ex lim f(x)
2 x?2 but, lim
f(x) 7 x?5
(not f(5) 1) and lim f(x) DNE
x?16 (DNE does not exist)
2
5
10
15
When we look at the limit below, we examine the
f(x) values as x gets very close to a
lim f(x)
x?a
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Example 1

• Answer each using the graph to the right (from
  Study Guide that accompanies Single Variable
  Calculus by Stewart)
•  
•  

Lim f(x) x? -5
4
Lim f(x) x? 2
3
Lim f(x) x? 0
DNE
Lim f(x) x? 4
0
11
Example 2
sin x Lim ------------
x
Use tables to estimate
x? 0
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Example 3
Use algebra to find a. b. c.
x³ - 1 Lim ------------
x - 1
Lim (x² x 1) 3 x? 1
x? 1
x - 1 Lim ------------
?x - 1
Lim (?x 1) 2 x? 1
x? 1
x 1 Lim ---------
- -------- x 1
x 1
Lim 1 1 x? 1
x? 1
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Summary Homework

• Summary
• Try to find the limit via direct substitution
• Use algebra to simplify into useable form
• Homework pg 102-104 5, 6, 7, 9