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1.2:Rates of Change & Limits
Learning Goals:
©2009 Mark Pickering
•Calculate average & instantaneous speed•Define, calculate & apply properties of limits•Use Sandwich Theorem
Important Ideas•Limits are what make calculus different from algebra and trigonometry•Limits are fundamental to the study of calculus•Limits are related to rate of change•Rate of change is important in engineering & technology
Theorem 1Limits have the following properties:
lim ( )x c
g x M
lim ( )x c
f x L
if &
then:
lim ( ) ( )x c
f x g x L M
1.
Theorem 1Limits have the following properties:
lim ( )x c
g x M
lim ( )x c
f x L
if &
then:
lim ( ) ( )x c
f x g x L M
2.
Theorem 1Limits have the following properties:
lim ( )x c
g x M
lim ( )x c
f x L
if &
then:
lim ( ) ( )x c
f x g x L M
3.
Theorem 1Limits have the following properties:
lim ( )x c
f x L
if
then:
lim ( )x c
k f x k L
4.
& k a constant
Theorem 1Limits have the following properties:
lim ( )x c
g x M
lim ( )x c
f x L
if &
then:( )
lim , 0( )x c
f x LM
g x M 5.
Theorem 1Limits have the following properties:
lim ( )x c
f x L
if &
6.
r & s are
integers, then:
lim ( )x c
rrssf x L
Theorem 1Limits have the following properties:
if where k is a
7.constant, then:lim ( ) lim
x c x cf x k k
( )f x k
(not in your text as Th. 1)
Theorem 2For polynomial and rational functions:
lim ( ) ( )x c
f x f c
( ) ( )lim , ( ) 0
( ) ( )x c
f x f cg c
g x g c
a.b.
Limits may be found by substitution
ExampleSometimes limits do not exist. Consider:
3
2
3lim
2x
x
x
If substitution gives a constant divided by 0, the limit does not exist (DNE)
Example
Find the limit if it exists:3
1
1lim
1x
x
x
Substitution doesn’t work…does this mean the limit doesn’t exist?
Procedure1.Try substitution2. Factor and cancel if
substitution doesn’t work
3.Try substitution again
The factor & cancellation technique
Try This
Find the limit if it exists:2
3
6lim
3x
x x
x
5
Isn’t
that
easy?
Did you think ca
lculus
was going to
be
difficu
lt?
Important IdeaThe limit of an indeterminate form exists, but to find it you must use a technique, such as factor and cancel.
Try This
Graph and
3
1
1
1
xY
x
2
2 1Y x x on the same axes. What is the
difference between these graphs?
Important Idea
The existence or non-existence of f(x) as x approaches c has no bearing on the existence of the limit of f(x) as x approaches c.
Important Idea
What matters is…what value does f(x) get very, very close to as x gets very,very close to c. This value is the limit.
Try ThisFind the limit of f(x) as x approaches 3 where f is defined by:
2 , 3( )
3 , 3
xf x
x
3lim ( ) 2x
f x
Theorem 3: One-sided & Two Sided limits
if lim ( )x c
f x L
(limit from right)
andlim ( )x c
f x L
(limit from left)
then lim ( )x c
f x L
(overall limit)
Theorem 3: One-sided & Two Sided limits
(Converse)if lim ( )
x cf x L
(limit from
right)andlim ( )x c
f x M
(limit from left)
then lim ( )x c
f x
(DNE)
Example
Consider
3 1( ) , 1
1
xf x x
x
What happens at x=1?
x .75 .9 .99 .999
f(x)
Let x get close to 1 from the left:
Try This
Consider
3 1( ) , 1
1
xf x x
x
x 1.25 1.1 1.01
1.001
f(x)
Let x get close to 1 from the right:
Try ThisWhat number does f(x) approach as x approaches 1 from the left and from the right?
3
1
1lim 3
1x
x
x
Important Idea
If f(x) bounces from one value to another (oscillates) as x approaches c, the limit of f(x) does not exist at c:
Theorem 4: Sandwich (Squeeze) Theorem
Let f(x) be between g(x) & h(x) in an interval containing c. Iflim ( ) lim ( )
x c x cg x h x L
lim ( )x c
f x L
then:
f(x) is “squeezed” to L
Example
Find the limit if it exists:
0
sin(5 )limx
x
x
0 0
sin(5 ) sin(5 )lim 5 5 lim 5 1 5
5 5x x
x x
x x