Level 1 and 2 Limits Graphically and at infinity.notebook 1 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 self satisfaction 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
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Level 1 and 2 Limits Graphically and at infinity.notebook
1
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
2
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