warm up packet pg. 2 & 3. problems for hw credit pg. 65 3: morgan; gil 39: victoria; rebecca 40:...
TRANSCRIPT
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