Math Tutoring TBA Please note: You will be assigned “Mandatory Math Tutoring” if you miss 3 assignments Calculus Tutoring Tuesday – 3-4 pm rm. 655 Thursday.

Math TutoringTBAPlease note: You will be assigned “Mandatory Math Tutoring” if you miss 3 assignments

Calculus TutoringTuesday – 3-4 pm rm. 655Thursday – 3-4 pm rm. 680

Get a copy of the reviewGet a copy of the review

-3

∞ *DNE?

-∞ *DNE?

DNE

Undefined

0

0

0

0

2

2

2

-1

0

0

0

undefined

-2

3

DNE

3

0

Calc

ula

tor

tim

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alc

ula

tor

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2.2 (CONT) Limits: 2.2 (CONT) Limits: numerically numerically Graphically estimate the limit of:Graphically estimate the limit of:

2

1lim 3 2x

x x

Estimate the limit using a table of values.Estimate the limit using a table of values.

Limits: numerically Limits: numerically Graphically estimate the limit of:Graphically estimate the limit of:

3

1

1lim

1x

x

x




21

1lim .

1x

xx

ExampleExample(cont’d)(cont’d)



2

20

9 3lim .t

tt

Cool fogCool fog

COOL FoGCOOL FoG

1.1. 00

2.2. 00

3.3. 11

4.4. 55

5.5. 11

6.6. 11

7.7. 11

8.8. 00

9.9. 22

10.10.11

11.11.11

12.12.11

Section 2.3Section 2.3

Calculating Limits Calculating Limits Algebraically Algebraically

SWBATSWBAT– Calculate limits using limit lawsCalculate limits using limit laws

Direct Substitution PropertyDirect Substitution Property

Let’s start with the Let’s start with the following:following:

If you can evaluate the limit at a, then do it.If you can evaluate the limit at a, then do it.This is also known as “This is also known as “Plug and ChugPlug and Chug””

lim ( ) ( )x a

f x f a

2

5 2x – 3x 4lim

x

If you cant “plug and chug”,If you cant “plug and chug”,Try something algebraic. Try something algebraic.

FindFind2

1

1lim

1x

xx

SolutionSolution

Here is a general principle:Here is a general principle:

2

1 1

1 11lim lim

1 1x x

x xxx x

1

lim 1x

x 1 1 2

more examplesmore examples2

24

5 4lim

3 4x

x x

x x

2

24

4lim

3 4x

x x

x x

2

0

(5 ) 25limx

x

x

ReviewReview

Using the Limit Laws to calculate Using the Limit Laws to calculate limitslimits

Additional Properties of LimitsAdditional Properties of Limits Direct Substitution PropertyDirect Substitution Property

Assignment 6Assignment 6

p. 106 1, 9-23 p. 106 1, 9-23 odd odd

Lets look at :Lets look at :

•You must do some algebra You must do some algebra or trig. To simplify first:or trig. To simplify first:

What happens when you evaluate this next function?What happens when you evaluate this next function?

You get what is called an You get what is called an Indeterminate formIndeterminate form (it can not be determined) like (it can not be determined) like 00//00

•Now plug and chug!:Now plug and chug!:•The limit is 3The limit is 3

3

1

1

1limx

x

x

2

1

( 1)( 1)

1limx

x x x

x

2

1( 1)lim

xx x

The Limit LawsThe Limit Laws(don’t write this down)(don’t write this down)

2( ) lim 5

xa f x g x

1( ) lim

xb f x g x

2

( ) limx

g xc

f x

4

. . .DNE

zero

Further Limit PropertiesFurther Limit Properties(if you think it would help you, then writ it down, if (if you think it would help you, then writ it down, if

not they are on p. 109)not they are on p. 109)

Applying the Product Law repeatedly Applying the Product Law repeatedly with with gg((xx) = ) = ff((xx) gives the following ) gives the following Power LawPower Law::

Here are two obvious but useful Here are two obvious but useful limits:limits:

Further Properties (cont’d)Further Properties (cont’d)

more examplesmore examples

2

lim ( ) ( )x

f x g x

1

lim ( ) ( )x

f x g x

0

lim ( ) ( )x

f x g x

1

( )lim

( )x

f x

g x

3

2lim ( )x

x f x

1lim 3 ( )x

f x

2.5 DNE 0

.588 ish 16 2

Math Tutoring TBA Please note: You will be assigned “Mandatory Math Tutoring” if you miss 3 assignments Calculus Tutoring Tuesday – 3-4 pm rm. 655 Thursday.

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