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)...
TRANSCRIPT
![Page 1: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/1.jpg)
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!
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If we increase the number of sides of the polygon, what can you say about the polygon
with respect to the circle?
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What is a limit?
Limit is the value of Y as X approaches a given #:
Lxfcx
)(lim
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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
![Page 5: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/5.jpg)
When do limits not exist? DNE
Video
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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
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Example 1: Find the following limits
![Page 8: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/8.jpg)
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1
1-
1
0
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0
0-
0
-3
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-2
-2-
-2
2
![Page 13: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/13.jpg)
Grab a graphing board, marker, and towel
![Page 14: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/14.jpg)
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![Page 20: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/20.jpg)
Limit Properties
These are important!
![Page 21: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/21.jpg)
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
![Page 22: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/22.jpg)
Practice:
11
7
3
2
1. lim 8
2. lim
3. lim
x
x
x
x
x
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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
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Practice1.
2.
![Page 25: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/25.jpg)
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
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x .75 .9 .99 .999 1 1.001 1.01 1.1 1.25
f(x) ?
1,1
1)(
3
xx
xxf
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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
![Page 28: 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](https://reader034.vdocuments.us/reader034/viewer/2022051215/5697bf771a28abf838c81343/html5/thumbnails/28.jpg)
Example: Graph )3)(2(
2
xx
xy
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Homework:
pg. 65 (1 – 4, 37 – 48, 79-82)Packet pg. 1