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Pop QuizUsing f & g, graphically find the limits for #1-6.

x 13. lim f(x)

x 16. lim f(x)g(x)

x 12. lim f(x)

x 0

g(x)5.lim

f(x)x 11. lim f(x)

x 04.lim 2f(x) 3g(x)

f(x)

g(x)

Pop QuizAlgebraically, find the limit of each in #7-10

3

h 0

h 1 19. lim

h

2

3t 2

t 48.lim

t 8

2

2x 3

x 97. lim

x 2x 3

x 010.lim cos(x)

Answers

1. 22. 13. DNE4. 45. 06. DNE

7. 3/28. 1/39. 310.1

Section 2.3

Calculating Limits Using the Limit Laws

SWBAT– Calculate limits using limit laws

Direct Substitution Property(Review)

Let’s start with the following:

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

lim ( ) ( )x a

f x f a

2

5 2x – 3x 4lim

x

Simplify with algebra:

Remember this?

This is called an Indeterminate form (it can not be determined)

• Now plug and chug!:• The limit is 3

3

1

1

1limx

x

x

2

1

( 1)( 1)

1limx

x x x

x

2

1( 1)lim

xx x

Plug and chug. You get 0/0

The Limit Laws(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 Properties(if you think it would help you, then writ it down, if

not they are on p. 109)

Applying the Product Law repeatedly with g(x) = f(x) gives the following Power Law:

Here are two obvious but useful limits:

Further Properties (cont’d)

more 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

WHEN FACTORING DOES NOT WORK 2

20

9 3limt

tt

1 1

3 3 6

0 2 2

0

1 1lim

9 3 lim 9 3t

tt t

2 2

2 20

9 3 9 3lim

9 3t

t tt t

2 2

0 02 2 2 2

9 9lim lim

9 3 9 3t t

t t

t t t t

MORE EXAMPLES

0

4 2limx

xx

3

30

4 16limh

hh

To review:A 2nd algebraic technique is

multiplying by a conjugate

0lim

25 5x

x

x

GREATEST INTEGER FUNCTION

Think of the number of teachers hired with respect to students enrolled.

EXAMPLES

_2

1. limx

x

2

2. limx

x

4.7

3. limx

x

Review

Using the Limit Laws to calculate limits

Additional Properties of Limits Direct Substitution Property

Assignment 7

p. 115 1-23 odd, 37