Objectives:1. Be able to find the limit of a function using direct substitution.2. Be able to find the limit of function that is in the indeterminate
form.
Critical Vocabulary:Limit, Direct substitution, Indeterminate Form
Warm Up: Find the limit numerically and graphically
3
372lim
2
3
x
xxx
Warm Up: Find the limit numerically and graphically
3
372lim
2
3
x
xxx
x
f(x)
-5
-9
-4
-7
-3.1
-5.2
-3.01
-5.02
-3.001
-5.002
-1
-1
-2
-3
-2.9
-4.8
-2.99
-4.98
-2.999
-4.998
-3
?
53
372lim
2
3
x
xxx
I. Direct Substitution
Example 1: Find the limit
1lim 2
1
x
xf(1) = (1)2 + 1 f(1) = 1 + 1
f(1) = 2
21lim 2
1
x
x
Example 2: Find the limit 3
4lim
2
2
x
xx
32
4)2()2(
2
f
1
0)2(
f
0)2( f
03
4lim
2
2
x
xx
I. Direct Substitution
Example 3: Find the limit 32
1lim
2
2
3
xx
x
x
323
23
2
123
2
32
f
323
49
2
123
2
3
f
323
29
123
2
3
f
321
2
3
f
6
1
2
3
f
6
1
32
1lim
2
2
3
xx
x
x
I. Direct Substitution
Example 4: Find the limit 1
32lim
2
1
x
xxx
11
31)1(2)1(
2
f
11
312)1(
f
0
0)1( f
Direct Substitution will not work on this because it yields an indeterminate form.
Undefined
Section 3.3 Assignment Part 1Page 226-228 #1-25 odd
MUST SHOW YOUR WORK TO GET CREDIT!!!!
II. Indeterminate Form
1. Factor and Cancel
2. Rationalizing Technique (Use when you have a radical)
3. Use Algebra (Common Denominators/Expand)
Then try to use direct substitution a second time!!!!!!
Example 4: Find the limit 1
32lim
2
1
x
xxx
1
)1)(32(lim
1
x
xxx
32lim1
xx
f(-1) = 2(-1) - 3 f(-1) = -5
532lim1
xx
Causes a hole in the graph
or an asymptote
This graph will have a hole at
(-1, -5)
II. Indeterminate Form
1. Factor and Cancel
2. Rationalizing Technique (Use when you have a radical)
3. Use Algebra (Common Denominators/Expand)
Then try to use direct substitution a second time!!!!!!
Example 5: Find the limit xx
xx 20lim
)1(lim
0 xx
xx
1
1lim
0 xx
f(0) = 1/(0-1)
f(0) = -1
1lim20
xx
xx
Causes a hole in the graph
or an asymptote
This graph will have a hole at
(0, -1)
II. Indeterminate Form
1. Factor and Cancel
2. Rationalizing Technique (Use when you have a radical)
3. Use Algebra (Common Denominators/Expand)
Then try to use direct substitution a second time!!!!!!
Example 6: Find the limit
x
xx
22lim
0
22
22
x
x
22lim
0
xx
xx
22
1lim
0
xx
202
1)0(
f
22
1
4
2
4
222lim
0
x
xx
Section 3.3 Assignment Part 2Page 226-228 #31-49 odd, 57
MUST SHOW YOUR WORK TO GET CREDIT!!!!
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