1.2:Rates of Change & Limits Learning Goals: © 2009 Mark Pickering Calculate average & instantaneous speed Define, calculate & apply properties of limits.

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.


lim ( )x c

g x M

lim ( )x c

f x L

if &

then:

lim ( ) ( )x c

f x g x L M

2.


lim ( )x c

g x M

lim ( )x c

f x L

if &

then:

lim ( ) ( )x c

f x g x L M

3.


lim ( )x c

f x L

if

then:

lim ( )x c

k f x k L

4.

& k a constant


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.


lim ( )x c

f x L

if &

6.

r & s are

integers, then:

lim ( )x c

rrssf x L


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

ExampleSolve using limit properties and substitution:

2

3lim 2 3 2x

x x

Try ThisSolve using limit properties and substitution:2

2

4lim

3x

x x

x

6

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)

ExampleTrig functions may have limits.

2

lim(sin )x

x

Try This

2

lim(cos )x

x

2

lim(cos ) cos 02x

x

Example

Find the limit if it exists:3

1

1lim

1x

x

x

Try substitution

Example


1

1lim

1x

x

x

Substitution doesn’t work…does this mean the limit doesn’t exist?

Important Idea3 21 ( 1)( 1)

1 1

x x x x

x x

2 1x x and

are the same except at x=-1

Important Idea

The functions have the same limit as x-1

Procedure1.Try substitution2. Factor and cancel if

substitution doesn’t work

3.Try substitution again

The factor & cancellation technique

Try This


3

6lim

3x

x x

x

5

Isn’t

that

easy?

Did you think ca

lculus

was going to

be

difficu

lt?

Try ThisFind the limit if it exists:

22

2lim

4x

x

x

1

4

Try This


3

6lim

3x

x x

x

The limit doesn’t existConfirm by graphing

DefinitionWhen substitution results in a 0/0 fraction, the result is called an indeterminate form.

Important IdeaThe limit of an indeterminate form exists, but to find it you must use a technique, such as factor and cancel.


2

1

2 3lim

1x

x x

x

-5

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?

3 1( )

1

xf x

x

Why is there a “hole” in the graph at x=1?

Analysis

ExampleConsider3 1

( )1

xf x

x

for ( ,1) (1, ) and

( ) 4f x

for x=1

3

1

1lim

1x

x

x

=?

Try ThisFind: if

1lim ( )x

f x

2( ) 2, 1f x x x

( ) 1, 1f x x

1lim ( ) 3x

f x

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 This

Find:

f(0)is undefined; 2 is the limit

2( )

1 1

xf x

x

0lim ( )x

f x

Find:

( ) 1, 0f x x

Try This

( ) , 01 1

xf x x

x

f(0) is defined; 2 is the limit

21

0lim ( )x

f x

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

Try ThisGraph and find the limit (if it exists):

3

3lim

3x x DNE

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


0limx

x

x

DNE

Example

Find the limit if it exists:

0

1lim sinx x

Example

1.Graph using a

friendly window:

1sin

x

2. Zoom at x=0

3. Wassup at x=0?

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

ExampleFind the limit if it exists:

0

sinlim

Where is in radians and in the interval,2 2


0

sinlim

Substitution gives the indeterminate form…


0

sinlim

Factor and cancel doesn’t work…


0

sinlim

Maybe…the squeeze theorem…

Example

g()=1

h()=cos

sin( )f

Example

0lim1 1

0

lim cos 1

&

therefore…

0

sinlim 1

Two Special Trig Limits

0

sinlim 1

0

1 coslim 0

Memoriz

e

Example


0

tanlimx

x

x

0 0

sin 1lim lim 1 1 1

cosx x

x

x x

Example


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

Try This


0

3 3 coslimx

x

x

0

Lesson Close

Name 3 ways a limit may fail to exist.

Practice

1. Sec 1.2 #1, 3, 8, 9-18, 28-38E (just find limit L), 39-42gc (graphing calculator), 43-45

1.2:Rates of Change & Limits Learning Goals: © 2009 Mark Pickering Calculate average & instantaneous speed Define, calculate & apply properties of limits.

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