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Copyright © 2011 Pearson Education, Inc. Slide 12.1-1
Limits
An Introduction To Limits
Techniques for Calculating Limits
One-Sided Limits; Limits Involving Infinity
Copyright © 2011 Pearson Education, Inc. Slide 12.1-2
Limit of a Function
The function
is not defined at x = 2, so its graph has a “hole” at x = 2.
2 4( )
2
xf x
x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-3
Limit of a Function
Values of may be computed near x = 2
x approaches 2
f(x) approaches 4
2 4( )
2
xf x
x
x 1.9 1.99 1.999 2.001 2.01 2.1
f(x) 3.9 3.99 3.999 4.001 4.01 4.1
Copyright © 2011 Pearson Education, Inc. Slide 12.1-4
Limit of a Function
The values of f(x) get closer and closer to 4 as x gets closer and closer to 2.
We say that
“the limit of as x approaches 2 equals 4”
and write
2 4
2
x
x
2
2
4lim 4.
2x
x
x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-5
Limit of a Function
Limit of a Function
Let f be a function and let a and L be real numbers. L is the limit of f(x) as x approaches a, written
if the following conditions are met.
1. As x assumes values closer and closer (but not equal ) to a on both sides of a, the corresponding values of f(x) get closer and closer (and are perhaps equal) to L.
2. The value of f(x) can be made as close to L as desired by taking values of x arbitrarily close to a.
lim ( ) ,x a
f x L
Copyright © 2011 Pearson Education, Inc. Slide 12.1-6
Finding the Limit of a Polynomial Function
Example Find
Solution The behavior of near x = 1 can be determined from a table of values,
x approaches 1
f(x) approaches 2
2
1lim ( 3 4).x
x x
x .9 .99 .999 1.001 1.01 1.1
f(x) 2.11 2.0101 2.001 1.999 1.9901 1.91
2( ) 3 4f x x x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-7
Finding the Limit of a Polynomial Function
Solution or from a graph of f(x).
We see that 2
1lim ( 3 4) 2.x
x x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-8
Finding the Limit of a Polynomial Function
Example Find where
Solution Create a graph and table.
3lim ( )x
f x
2 1 if 3( ) .
4 5 if 3
x xf x
x x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-9
Finding the Limit of a Polynomial Function
Solution x approaches 3
f(x) approaches 7
Therefore3
lim ( ) 7.x
f x
x 2.9 2.99 2.999 3.001 3.01 3.1
f(x) 6.8 6.98 6.998 7.004 7.04 7.4
Copyright © 2011 Pearson Education, Inc. Slide 12.1-10
Limits That Do Not Exist
• If there is no single value that is approached
by f(x) as x approaches a, we say that f(x)
does not have a limit as x approaches a,
or does not exist. 2
lim ( )x
f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-11
Determining Whether a Limit Exists
Example Find where
Solution Construct a table and graph
x 1.9 1.99 1.999 2.001 2.01 2.1
f(x) 2.6 2.96 2.996 1.003 1.03 1.3
2lim ( )x
f x
4 5 if 2( ) .
3 5 if 2
x xf x
x x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-12
Determining Whether a Limit Exists
Solution
f(x) approaches 3 as x gets closer to 2 from the left,f(x) approaches 1 as x gets closer to 2 from the right.
Therefore, does not exist.2
lim ( )x
f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-13
Determining Whether a Limit Exists
Example Find where
Solution Construct a table and graph
0lim ( )x
f x 2
1( ) .f x
x
x -.1 -.01 -.001
f(x) 100 10,000 1,000,000
x .001 .01 .1
f(x) 1,000,000 10,000 100
Copyright © 2011 Pearson Education, Inc. Slide 12.1-14
Determining Whether a Limit Exists
Solution
As x approaches 0, the corresponding values of f(x) grow arbitrarily large.
Therefore, does not exist.20
1limx x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-15
Limit of a Function
Conditions Under Which Fails To Exist
1. f(x) approaches a number L as x approaches a from the left and f(x) approaches a different number M as x approaches a from the right.
2. f(x) becomes infinitely large in absolute value as x approaches a from either side.
3. f(x) oscillates infinitely many times between two fixed values as x approaches a.
lim ( )x a
f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-16
Limits
1. An Introduction To Limits
2. Techniques for Calculating Limits
3. One-Sided Limits; Limits Involving Infinity
Copyright © 2011 Pearson Education, Inc. Slide 12.1-17
Techniques For Calculating Limits
Rules for Limits
1. Constant rule If k is a constant real number,
2. Limit of x rule For the following rules, we assume that and
both exist
3. Sum and difference rules
lim .x a
k k
lim .x a
x a
lim[ ( ) ( )] lim ( ) lim ( ).x a x a x a
f x g x f x g x
lim ( )x a
f x
lim ( )x a
g x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-18
Techniques For Calculating Limits
Rules for Limits
4. Product Rule
5. Quotient Rule
provided
lim[ ( ) ( )] lim ( ) lim ( ).x a x a x a
f x g x f x g x
lim ( )( )lim .
( ) lim ( )x a
x ax a
f xf x
g x g x
lim ( ) 0.x a
g x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-19
Finding a Limit of a Linear Function
Example Find
Solution
Rules 1 and 4
Rules 1 and 2
4lim (3 2 ).x
x
4 4 4lim (3 2 ) lim 3 lim 2x x x
x x
4 43 lim 2 lim
x xx
3 2 4
11
Copyright © 2011 Pearson Education, Inc. Slide 12.1-20
Finding a Limit of a Polynomial Function with One Term
Example Find
Solution Rule 4
Rule 1
Rule 4
Rule 2
2
5lim 3 .x
x
2 2
5 5 5lim 3 lim 3 limx x x
x x
2
53 lim
xx
5 53 lim lim
x xx x
3 5 5
75
Copyright © 2011 Pearson Education, Inc. Slide 12.1-21
Finding a Limit of a Polynomial Function with One Term
For any polynomial function in the form ( ) ,nf x kx
lim ( ) ( ).n
x af x k a f a
Copyright © 2011 Pearson Education, Inc. Slide 12.1-22
Finding a Limit of a Polynomial Function
Example Find .
Solution
Rule 3
3
2lim (4 6 1)x
x x
3 3
2 2 2 2lim (4 6 1) lim 4 lim 6 lim 1x x x x
x x x x
34 2 6 2 1
21
Copyright © 2011 Pearson Education, Inc. Slide 12.1-23
Techniques For Calculating Limits
Rules for Limits (Continued)
For the following rules, we assume that and
both exist.
6. Polynomial rule If p(x) defines a polynomial function, then
lim ( )x a
f x
lim ( )x a
g x
lim ( ) ( ).x a
p x p a
Copyright © 2011 Pearson Education, Inc. Slide 12.1-24
Techniques For Calculating Limits
Rules for Limits (Continued)
7. Rational function rule If f(x) defines a rational
function with then
8. Equal functions rule If f(x) = g(x) for all , then
lim ( ) ( ).x a
f x f a
( )
( )
p x
q x( ) 0q a
x a
lim ( ) lim ( ).x a x a
f x g x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-25
Techniques For Calculating Limits
Rules for Limits (Continued)
9. Power rule For any real number k,
provided this limit exists.
lim[ ( )] lim ( )k
k
x a x af x f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-26
Techniques For Calculating Limits
Rules for Limits (Continued)
10. Exponent rule For any real number b > 0,
11. Logarithm rule For any real number b > 0 with ,
provided that
lim ( )( )lim .x af xf x
x ab b
1b
lim log ( ) log lim ( )b bx a x a
f x f x
lim ( ) 0.x a
f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-27
Finding a Limit of a Rational Function
Example Find
Solution Rule 7 cannot be applied directly since the denominator is 0. First factor the numerator and denominator
2
21
2 3lim .
3 2x
x x
x x
2
2
2 3 ( 3)( 1) 3
3 2 ( 2)( 1) 2
x x x x x
x x x x x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-28
Finding a Limit of a Rational Function
Solution Now apply Rule 8 with
and
so that f(x) = g(x) for all .
2
2
2 3( )
3 2
x xf x
x x
3
( )2
xg x
x
1x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-29
Finding a Limit of a Rational Function
Solution Rule 8
Rule 6
2
21 1
2 3 3lim lim
3 2 2x x
x x x
x x x
1 3
1 2
4
Copyright © 2011 Pearson Education, Inc. Slide 12.1-30
Limits
1 An Introduction To Limits
2 Techniques for Calculating Limits
3 One-Sided Limits; Limits Involving Infinity
Copyright © 2011 Pearson Education, Inc. Slide 12.1-31
One-Sided Limits
Limits of the form
are called two-sided limits since the values of x get close to a from both the right and left sides of a.
Limits which consider values of x on only oneside of a are called one-sided limits.
lim ( )x a
f x L
Copyright © 2011 Pearson Education, Inc. Slide 12.1-32
One-Sided Limits
The right-hand limit,
is read “the limit of f(x) as x approaches a from the right is L.”
As x gets closer and closer to a from the right (x > a), the values of f(x) get closer and closer to L.
lim ( )x a
f x L
Copyright © 2011 Pearson Education, Inc. Slide 12.1-33
One-Sided Limits
The left-hand limit,
is read “the limit of f(x) as x approaches a from the left is L.”
As x gets closer and closer to a from the right (x < a), the values of f(x) get closer and closer to L.
lim ( )x a
f x L
Copyright © 2011 Pearson Education, Inc. Slide 12.1-34
Finding One-Sided Limits of a Piecewise-Defined Function
Example Find and where
2lim ( )x
f x
2
6 if 2
5 if 2( )
1if 2
2
x x
xf x
x x
2lim ( )x
f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-35
Finding One-Sided Limits of a Piecewise-Defined Function
Solution Since x > 2 in use the formula
. In the limit , where x < 2, use
f(x) = x + 6.
2lim ( )x
f x
2lim ( )x
f x
2 2
2 2
2 2
1 1lim ( ) lim 2 2
2 2
lim ( ) lim ( 6) 2 6 8
x x
x x
f x x
f x x
21( )
2f x x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-36
Infinity as a Limit
A function may increase without bound as x gets closer and closer to a from the right
Copyright © 2011 Pearson Education, Inc. Slide 12.1-37
Infinity as a Limit
The right-hand limit does not exist but the behavior is described by writing
If the values of f(x) decrease without bound, write
The notation is similar for left-handed limits.
lim ( )x a
f x
lim ( )x a
f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-38
Infinity as a Limit
Summary of infinite limits
Copyright © 2011 Pearson Education, Inc. Slide 12.1-39
Finding One-Sided Limits
Example Find and where
Solution From the graph
2lim ( )x
f x
1( ) .
2f x
x
2lim ( )x
f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-40
Finding One-Sided Limits
Solution and the table
and2
lim ( )x
f x
2
lim ( ) .x
f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-41
Limits as x Approaches +
A function may approach an asymptotic value as
x moves in the positive or negative direction.
lim ( ) 2x
f x
lim ( ) 1x
g x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-42
Limits as x Approaches +
The notation,
is read “the limit of f(x) as x approaches infinity is L.”
The values of f(x) get closer and closer to L as x gets larger and larger.
lim ( )x
f x L
Copyright © 2011 Pearson Education, Inc. Slide 12.1-43
Limits as x Approaches +
The notation,
is read “the limit of f(x) as x approaches negative infinity is L.”
The values of f(x) get closer and closer to L as x assumes negative values of larger and larger magnitude.
lim ( )x
f x L
Copyright © 2011 Pearson Education, Inc. Slide 12.1-44
Finding Limits at Infinity
Example Find and where
Solution As the values of e-.25x get arbitrarily close to 0 so
lim ( )x
f x
.25
10( ) 5 .
1 xf x
e
lim ( )x
f x
x
10lim ( ) 5 15.
1 0xf x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-45
Finding Limits at Infinity
Solution As the values of e-.25x get arbitrarily large so
x
lim ( ) 5 0 5.x
f x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-46
Finding Limits at Infinity
Solution (Graphing calculator)
Copyright © 2011 Pearson Education, Inc. Slide 12.1-47
Limits as x Approaches +
Limits at infinity of
For any positive real number n,
and1
lim 0nx x
1nx
1lim 0.
nx x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-48
Finding a Limit at Infinity
Example Find
Solution Divide numerator and denominator by the highest power of x involved, x2.
2
2
5 7 1lim .
2 5x
x x
x x
2 2
2
2
7 155 7 1
lim lim1 52 5 2
x x
x x x xx x
x x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-49
Finding a Limit at Infinity
Solution 2 2
2
2
2
2
7 155 7 1
lim lim1 52 5 2
7 1lim 5
1 5lim 2
x x
x
x
x x x xx x
x x
x x
x x
Copyright © 2011 Pearson Education, Inc. Slide 12.1-50
Finding a Limit at Infinity
Solution
2 2
2
2
1 1lim 5 7 lim lim5 7 1
lim1 12 5 lim 2 lim 5 lim
5 0 0 5
2 0 0 2
x x x
x
x x x
x x x xx x
x x
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