Warm Up: No Calc
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Warm Up: No Calc 1. Find all asymptotes for (A) x=1, x=-1, y=1 (B) x=1, y=1 (C) x=1, x=-1, y=0 (D) x=1, x=-1 (E) y=1 2. 3. Use properties of logarithms to decide which of the following is largest. 2 2 () 1 x x fx x ln4 ( )ln(30) ln(2) ( )2ln4 ( )ln(3) ln(4) () ln2 A B C D Pick up new packet!
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Pick up new packet!. Warm Up: No Calc. 1. Find all asymptotes for (A) x=1, x=-1, y=1 (B) x=1, y=1(C) x=1, x=-1, y=0 (D) x=1, x=-1(E) y=1 2. 3. Use properties of logarithms to decide which of the following is largest. - PowerPoint PPT Presentation
Transcript of Warm Up: No Calc
Warm Up: No Calc1. Find all asymptotes for
(A) x=1, x=-1, y=1 (B) x=1, y=1 (C) x=1, x=-1, y=0 (D) x=1, x=-1 (E) y=1
2.
3. Use properties of logarithms to decide which of the following is largest.
2
2( )
1
x xf x
x
ln 4( ) ln(30) ln(2) ( )2 ln 4 ( ) ln(3) ln(4) ( )
ln 2A B C D
Pick up new packet!
If we increase the number of sides of the polygon, what can you say about the polygon
with respect to the circle?
What is a limit?
Limit is the value of Y as X approaches a given #:
Lxfcx
)(lim
3 Kinds of Limits:
Left – Hand Limit
As x approaches from the left side of c
Right – Hand Limit
As x approaches from the right side of c.
Limit (double – sided)
As X approaches c from either direction.
Only exists if left-hand and right-hand are the same.
Lxfcx
)(limLxfcx
)(limLxfcx
)(lim
When do limits not exist? DNE
Video
THM: Existence of a Limit
lim
lim
lim
x c
x c
x c
f x L iff
f x L AND
f x L
Example 1: Find the following limits
1
1-
1
0
0
0-
0
-3
-2
-2-
-2
2
Grab a graphing board, marker, and towel
Limit Properties
These are important!
Limits to KnowLet b & c be real numbers and let n be a positive integer.
1. The limit of a constant function is the constant.
2. The limit at any x-value on the line y=x IS the x-value itself.
3. The limit at any x-value of any function of the form y = xn is the x-value raised to the nth power.
limx cb b
limx cx c
limnn
x cx c
Practice:
11
7
3
2
1. lim 8
2. lim
3. lim
x
x
x
x
x
Properties of LimitsLet b & c be real # and n a positive int. and
lim ( ) lim ( )x c x cf x L g x K
1.lim[ * ] *x cb f x b L
2.lim[ ]x c
f x g x L K
Scalar multiple
Sum or Differ.
3.limx c
f x g x LK
4.lim , 0x c
f x LK
g x K
Product
Quotient
5.lim[ ]n n
x cf x L
Power
Practice1.
2.
Another nice thing about limits…• They help us find holes in the graph.• Ex: What will happen at x=1?
1,1
1)(
3
xx
xxf
x .75 .9 .99 .999 1 1.001 1.01 1.1 1.25
f(x) ?
1,1
1)(
3
xx
xxf
While f(1) D.N.E., as x moves arbitrarily close to 1 from the left and right, f(x)
moves arbitrarily close to 3.
“The limit of f(x) as x approaches 1 is 3”
3)(lim1
xfx
Example: Graph )3)(2(
2
xx
xy
Homework:
pg. 65 (1 – 4, 37 – 48, 79-82)Packet pg. 1
