Definition and finding the limit

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

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

Calculus Date: 9/26/14

Objective: SWBAT define, calculate & apply properties of limits graphically, numerically and now analytically.

Do Now –

Mini Quiz 5 minutes Take out a piece of paper. Can be a half sheet.

HW Requests: HW: pg 30 SM all

In class: worksheets ( possibilities Worksheet limits or McCafrey)

HW: Complete WorksheetsAnnouncements:

Mandatory session Sine and Cosine functions starting

with the Unit Circle

Quiz Friday

To get ahead,You have to do extra!

Mini Quiz7 minutes

2.Show your work then

1. Try Substitution, if doesn’t work

2. Try Factor and cancel and then3. Try Substitution again, if doesn’t work4. Do DNE or +/- infinity check

Techniques-Finding limits for Rational Expressions

Let’s go to the SM pg #28 #1-8 (10) HW: pg 30 SM all

- If the right and left side limit are not equal the limit does not exist - DNE

- If the right and left side limit are not equal the limit does not exist – DNE

One sided Limits- If the left side number is negative then the - If the left side number is positive then the - If the right side number is negative then the - If the left side number is negative then the

Rationalizing Technique

x

xx

11lim

0

If there is a radical in the numerator or the denominator,rationalize, simplify and cancel, then try substitution.

)11(

1

)11()11(

11

11

11.

11

x

xx

x

xx

x

x

x

x

x

Substituting we get 2

1

)110(

1

Hint: Often you can cancel a common term in the numerator and denominator when simplifying

Rationalizing Technique

xxx

x

x

x

x

x

2

1

)2)(4(

4

2

2.

4

2

Rationalize, simplify (cancel) and try substitution.

Substituting we get = 1 4

Try This

Find:

f(0)is undefined; 2 is the limit

2( )

1 1

xf x

x

0lim ( )x

f x

Find: Try This

( ) , 01 1

xf x x

x

f(0) is defined; 2 is the limit

21

0lim ( )x

f x

1, x = 0

Try ThisFind the limit if it exists:

0limx

x

x

DNE

Try ThisFind: if

1lim ( )x

f x

1lim ( ) 3x

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

Example

Find the limit if it exists:3

1

1lim

1x

x

x

Try substitution

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?

Use the factor & cancellation technique

3 21 ( 1)( 1)

1 1

x x x x

x x

2 1x x and

are the same except at x=-1

Use the factor & cancellation technique

2 1x x

After factoring and cancelling, now try substituting -1 again.

= 33

1

1lim

1x

x

x

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?

Try ThisSolve using limit properties and substitution:2

2

4lim

3x

x x

x

6

Try ThisFind the limit if it exists:

22

2lim

4x

x

x

1

4

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)

Try This

Find the limit if it exists:2

3

6lim

3x

x x

x

The limit doesn’t existConfirm by graphing

Lesson Close

Name 3 ways a limit may fail to exist.

Exit Ticket

• In Class: SM – pg 28 #1-5

• HW: SM pg 30 #1-15

Try ThisFind the limit if it exists:

2

1

2 3lim

1x

x x

x

-5

Limit properties again

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.

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.

In order for a limit to exist at c, the left-hand limit must equal the right hand limit.

Limits, again!

)(lim)(lim xfxfcxcx

Lxfcx

)(lim

If the left-hand limit equals the right hand limit, then the limit exists and we write:

Watch out for piecewise functions

When finding the limit of a function it is important to let x approach a from both the right

and left. If the same value of L is approached by the function then

the limit exist and

Lxf )(limax

ExampleConsider3 1

( )1

xf x

x

for ( ,1) (1, ) and

3

1

1lim

1x

x

x

=?

 

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

3

3lim

3x x DNE

ExampleTrig functions may have limits.

2

lim(sin )x

x

Try This

2

lim(cos )x

x

2

lim(cos ) cos 02x

x

Using the Product Rule Technique

 

Important Idea

The functions have the same limit as x-1

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

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

Informal Definition of Limit• If f(x) becomes arbitrarily close to a single

number L as x approaches a number c from either side, the limit of f(x), as x approaches c, is L.

Lxfcx

lim

- Definition of Limit

• Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement means that for each >0 there exists a >0 such that if then .

Lxfcx

lim

cx0 Lxf

Basic Limits

•

•

•

bbcx

lim

cxcx

lim

nn

cxcx

lim

Constant Function Limits

bbax

lim

• a and b are both constants

• This means that for any constant function f(x) = b, as x approaches any constant a, the limit will always be b.

Linear Function Limits

axax

lim

The limit of f(x) = x as x approaches any constant is the constant itself.

Exponential Function Limits

ax

axbb

lim

Just plug in a for x

Properties

Let and

Scalar multiple: Sum or difference:

Product:

Quotient: , if K0

Power:

Lxfcx

lim Kxgcx

lim

bLxbfcx

lim

KLxgxfcx

lim

LKxgxfcx

lim

K

L

xg

xfcx

lim

nn

cxLxf

lim

• Let's try a practice problem.

• Property (B) tells us we can split these apart:

• Using limit (1) and limit (2) from the basic limits, we get:

3lim2

xx

3limlim22

xx

x

1323lim

33lim

2lim

2

2

2

x

x

x

x

x

Putting it all together

• So,

5

714131

743lim

23

23

1

xxxx

This is called the Substitution method

Try ThisSolve using limit properties and substitution:

2

3lim 2 3 2x

x x

25