level 1 and 2 limits graphically and at...
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Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Warm-Up:
1.
2.
3.
How did you like doing a project versus a test?
include: time management creativity it allowed own work, but w/outside help Responsibility
Was learning on your own new to you?
In college you will have to read and learn on your own, did this experience give you a taste of this? Did this experience help you develop the skills to do this? or did it create an opportunity as a transitional step to developing these skills?
Pick one of the life skills below that you want to challenge yourself to develop before you leave math/graduate high school. Write 3 things you can do that will display your development:
• selfmotivation (getting yourself to do your best for selfsatisfaction that you grew or learned more),
• asking questions in class,
• selfdiscipline(getting yourself to do what needs to be done according to the situation whether you like it/want to or not),
• explaining your thought processes,
• homework completion/responsibility,
• staying awake,
• taking detailed/useful/color coordinated notes,
• participating as if the teacher was only teaching you
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
finding limits numerically
and graphically
Ch.15.1 Level 1
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Target
Agenda
Purpose
Evaluation
TSWBAT: Find the limit of a function given its graph or a table. Define and determine continuity
Warm-Up
Lesson
BAT: Develop desired life skill Get ready for calculus. Extend knowledge functions, make connections, apply knowledge to new situations
3-2-1
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Do we have limits?
Are there points where we are
physically bounded?
emotionally limited?
Or is it all in our minds?
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
As x approaches a, f(x) approaches L
Remember...
as x ⇒∞, y ⇒a
as x ⇒a, y ⇒∞
What if we aren't specifically looking at asymptotes?
Can the function pass y=3 as x keeps going to the left? What might you call y=3?
as x ⇒a, y ⇒L
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Limit Notation
lim f(x) = Lx ⇒ a
The values of f(x) (the y values) get closer to the number L as x gets closer and closer to the number a from either side of a, but x≠a.
As x approaches a, f(x) approaches L
as x ⇒a, y ⇒L
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
ex. What does f(x) approach to as x approaches 2
lim ( 2x2 2x + 4) =
Finding the Limit NumericallyUse your calculator
1. type in the function
2. Go to tblset (2nd Window) and adjust Tbl = .001 and TblStart = 1.998
X Y1.9971.9981.9992.0012.0022.003
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Cool Tips
F(x) doesn't have to be defined at a, just approach it from both sides
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Finding the Limit Graphically
lim f(x) =x⇒1
lim f(x) =x⇒3
lim f(x) =x⇒2
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
The limit is undefined!
3 Occurances : when the function is...
a Piecewise function with a jump at the point a that x approaches) :
lim f(x) =
lim f(x) =
lim f(x) =
an Oscillating function : f(x) = sin (π/x)
has a Vertical Asymptote : f(x) = 1/x2
x⇒1
x⇒2
x⇒1
lim f(x) =x⇒1
lim f(x) =x⇒0
lim f(x) =x⇒2
lim f(x) =x⇒∞
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
One Sided Limits
the limit of f(x) as x approaches 2 from the left
the limit of f(x) as x approaches 2 from the right
lim f(x) =x⇒2
lim f(x) =x⇒2+
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Does the limit exist? The Test
lim f(x) = L, iff lim f(x) = L = lim f(x)x⇒a x⇒a x⇒a+
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Continuity: is it continuous?In other words: do you have to lift up your pencil to draw it?
How a function can be continuous
1. f(a) must exist
2. lim f(x) must exist
3. They must be the same: f(a) = lim f(x)
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Continuity: These are not continuous
In other words: you do have to lift up your pencil to draw it?
How a function could fail
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
the limit of infinity
Ch.15.1 Level 2
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Target
Agenda
Purpose
Evaluation
TSWBAT: Find the limit as x approaches infinity
Warm-Up
Lesson
BAT: Get ready for calculus. Extend knowledge functions, make connections, apply knowledge to new situations
3-2-1
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
This is CRAZY!
What is the lim 3x2 4x + 2?
What is the lim x4 + x2 2 ? 44x
x⇒∞
x⇒∞
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
What is the limit as x approaches the infinities
lim f(x) = L or as x ⇒ ∞ , f(x) ⇒ Lx⇒∞
x⇒∞lim f(x) = L or as x ⇒ ∞ , f(x) ⇒ L
Negative Infinity: To the left, to the left!
Positive Infinity: To the right
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Can I hear an AsymptOTE!!!
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
When the denominator has greater degree then...
lim 1 xx⇒∞
as x gets bigger, 1/x gets smaller ⇒ 0, since ∞ isn't a number, we can get as close to 0 as we want
Special Infinity Rules
lim 1 xx⇒∞ k = 0 lim 1
xx⇒∞ k = 0
lim exx⇒∞ =0
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Special Infinity Rules Example
ex. f(x) = 4x3 x 3x3 + 2
ex. f(x) = 4x23 x 3x13 + 2x20
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
SummarizingTIME!
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014
Evaluation:1.
2.
Practice! Worksheet
lim f(x) =x⇒4
lim f(x) =x⇒2+
lim x⇒∞
10x 2 x3
3.
Level 1 and 2 Limits Graphically and at infinity.notebook
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May 07, 2014