Lesson 2-2 part 2

1
Lesson 2-2 part 2

• The Limit of a Function

2
Objectives

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

3
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

4
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 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) 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
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
•  
•  

Lim f(x) x? -2-
1
Lim f(x) x? -2
0
Lim f(x) x? -2
DNE
Lim f(x) x? 2-
3
Lim f(x) x? 2
0
Lim f(x) x? 2
DNE
Lim f(x) x? 0-
1
Lim f(x) x? 0
1
Lim f(x) x? 0
1
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Example 2
3 - x Lim ------------
x 3
Find
1
x? 3-
8
Example 3
Find a.
3x 1
x lt 2 Lim f(x) if f(x)
8 x 2
x² 3 x gt 2
Lim f(x) 7 x? 2
x? 2
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Example 4
Always, Sometimes or Never True a. If
does not exist, then
does not exist. b. If
does not exist, then
does not exist.
Lim f(x)
Lim f(x)
x? 2
x? 2
Sometimes --- if a two-sided limit is DNE,
then a one-sided limit might
be DNE
Lim f(x)
Lim f(x)
x? 2
x? 2
Always --- if a one-sided limit is DNE,
then the two-sided limit must
be DNE
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Example 5

• Answer each using the graph to the right
•  
•  

Lim f(x) x? -2-
DNE ( ?)
Lim f(x) x? -2
DNE ( ?)
Lim f(x) x? -2
DNE ( ?)
Lim f(x) x? 3-
DNE ( ?)
Lim f(x) x? 3
0
Lim f(x) x? 3
DNE only
Lim f(x) x? 0-
DNE (- ?)
Lim f(x) x? 0
DNE ( ?)
Lim f(x) x? 0
DNE only
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Example 6
True/False If
and
, then
.
Lim f(x) ? x? a
Lim g(x) ? x? a
False
Lim f(x) g(x) 0 x? a
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Summary Homework

• Summary
• Try to find the limit via direct substitution
• Use algebra to simplify into useable form
• Graph the function
• Homework pg 102-104 12, 19, 21, 23, 24, 27