Warm Up

Packet pg. 2 & 3

Problems for HW Credit

pg. 65

3: Morgan; Gil

39: Victoria; Rebecca

40: Megan; Drayton

44: Savona; Carlos

47: Decoya; Julie

48: Brenden; Brian

80: MacKenzie; LeKurt

Grab a large whiteboard and show us what you did…

It doesn’t have to be right but you have to show effort

Homework: pg. 65 (1 – 4, 37 – 48, 79-82)Packet pg. 1

Packet pg. 1

1. T: a, b, d, e, f

2. F: b, c

3. C

4. B

5. D

6. A

AP Calculus AB

Evaluating Limits Algebraically

DISCLAIMER:

The limit of f(x) as x approaches c DOES NOT depend on the value of f(c).

However, sometimes you get a gift and it’s a simple plug & chug problem when:

limx cf x f c

For these problems, simply sub in x = c.

i.e. DIRECT SUBSTITUTION

How to tell when you can use DIRECT SUBSTITUTION:

The function needs to be “well-behaved” (continuous) at c.

Which means (informally):

It is usually easy to try this in your head.

No holesNo jumpsNo asymptotesYou never have to pick up your pencil.

Thm:

If 2 functions agree at all but one point, then

lim ( ) lim ( )x c x cf x g x

So, what’s the big deal?...Helps us fill in the holes!!

Trick 1: Factoring

2 3 4( ) ( ) 1

4

x xf x g x x

x

f(-4) D.N.E., however

4 4lim ( ) lim ( ) 5x x

f x g x

Practice:

2

2

4( ) , lim ( )

2 x

xf x f x

x

3

0( ) , lim ( )

x

x xg x g x

x

3

1

1( ) , lim ( )

1 x

xh x h x

x

Note: When direct substitution produces 0/0, the expression is called Indeterminate Form…

0

1 1limx

x

x

Remember Conjugates?!?...

Trick 2: Rationalize the square root…

You try:

0

1

4 21. lim

12. lim

3 3

h

x

h

h

x

x

Trick 3: Multiply by 1 in a “convenient form” (The common denominator)

0

16

) lim1

3t

tex

t

0

1 12 2You try: lim

x

xx

One last way to find a limit…

Squeeze Thm (aka Sandwich Thm):

If

and

then

*Good for finding limits involving trig functions

( ) ( ) ( )h x f x g x

lim ( ) lim ( )x c x ch x L g x

lim ( )x cf x L

Big Picture

2

0

1) lim sinx

ex xx

Example 2:

Thm: KNOW THESE!! (hint!)

0

sin1. lim 1

x

x

x

0

1 cos2. lim 0

x

x

x

Examples:

1.

2.

Strategies for Finding Limits:

1. Go for the easy path 1st!! Try direct substitution2. Try to change the function into one that can be

solved by direct substitution (see previous slide)

3. Apply theorem to conclude that

4. Remember you can always graph to check!…But sometimes there is NO LIMIT.

Bwauh-haa-haa!!!

lim ( ) lim ( ) ( )x c x cf x g x g c

Limit Dominos