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COVID Year 8 limits, continuity, IVT.notebook
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September 30, 2020
What is a limit?
COVID Year 8 limits, continuity, IVT.notebook
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So according to the College Board students will know that...
Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is
Hmmm...I am puzzled by this! What about piecewise functions???
COVID Year 8 limits, continuity, IVT.notebook
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Infinite LimitsConsider the behavior of f(x) = 1/x for values of x near 0.
What do the following expressions mean?
and
and
Graphically what do these limits imply?
COVID Year 8 limits, continuity, IVT.notebook
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Limits at InfinityConsider the behavior of f(x) = 1/x when:
a) x increases without bound (x )
b) x decreases without bound (x )
Graphically what do these limits imply?
COVID Year 8 limits, continuity, IVT.notebook
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Famous Limit at InfinityLet
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COVID Year 8 limits, continuity, IVT.notebook
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What about basic trig functions?Consider the graph of f(x) = sin x
Evaluate:
COVID Year 8 limits, continuity, IVT.notebook
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When would limits not exist?
1)
2)
3)
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&uact=8&ved=0ahUKEwi776PUsP3OAhVD5SYKHbmQBCYQtwIIKzAE&url=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DoDAKKQuBtDo&usg=AFQjCNFRRblXa6oCPMmySwddC5Sf1Mhz2Q
COVID Year 8 limits, continuity, IVT.notebook
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COVID Year 8 limits, continuity, IVT.notebook
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COVID Year 8 limits, continuity, IVT.notebook
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Evaluate each by any means necessary!!!!
Find
Find
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All reals, x cannot equal 1
Line y = x 4, with a hole at x = 1
5
5
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5
The intended destination from left and right
COVID Year 8 limits, continuity, IVT.notebook
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Basic LimitsLet a and k represent real numbers.
1)
2)
3)
4)
COVID Year 8 limits, continuity, IVT.notebook
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Evaluate:
COVID Year 8 limits, continuity, IVT.notebook
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Classwork Page 67 #55
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Evaluate the following limit in two different ways,
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Homework
page 67 #56
page 76 #2, 13
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Limits of Polynomial Functions as x approaches ±∞Consider the graphs of each and exam the limits as x ±∞ a) y = x
b) y = x2
c) y = x3
d) y = x4
e) y = x2
What general results follow from these special cases?
COVID Year 8 limits, continuity, IVT.notebook
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Ex: Evaluate the following limits
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Limits of Piecewise FunctionsWe determine the limits of piecewise functions by considering the one sided limits.
Example:
Evaluate: (a)
(b)
(c)
First graph then evaluate.
COVID Year 8 limits, continuity, IVT.notebook
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Rational FunctionsFind
Find
Find
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Find
Theorem:Let be a rational function, and let a be any real number.
(a) If q(a) ≠ 0, then
(b) If q(a) = 0 but p(a) ≠ 0, then
(c) If q(a) = 0 = p(a), then
COVID Year 8 limits, continuity, IVT.notebook
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Rational Functions in which the limit of the denominator = 0and as x ±∞
COVID Year 8 limits, continuity, IVT.notebook
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Find
What do you think the limit will be?
COVID Year 8 limits, continuity, IVT.notebook
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Find What do you think?
COVID Year 8 limits, continuity, IVT.notebook
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Find What do you think?
COVID Year 8 limits, continuity, IVT.notebook
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What are these limits really finding?
COVID Year 8 limits, continuity, IVT.notebook
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Limits Involving RadicalsEvaluate What do you think?
COVID Year 8 limits, continuity, IVT.notebook
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Find
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7) Find
COVID Year 8 limits, continuity, IVT.notebook
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8) Find
COVID Year 8 limits, continuity, IVT.notebook
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9) Find
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Before we begin the next example let's review a fun fact, shall we?
=
Find
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10) Find
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Evaluate:
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4/3 0 2
8 12 0
4/5 4/3
COVID Year 8 limits, continuity, IVT.notebook
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COVID Year 8 limits, continuity, IVT.notebook
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2
DNE 1/4
k = 0
neg. infinity infinity
1.5 0
infinity
1
COVID Year 8 limits, continuity, IVT.notebook
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Limits of Exponential Functions
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Rookie Mistakes Practice1)
2)
3)
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End Behavior of Logarithmic Functions
COVID Year 8 limits, continuity, IVT.notebook
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Absolute ValueGiven, find
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2
2
1/4
2
1
2
1
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1/2
1/2
DNE
infinity
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Trigonometric Functions
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Memorization Card
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Example:
Evaluate
What do you think?
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Trig Limits
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Memorization Card
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What do you think?
COVID Year 8 limits, continuity, IVT.notebook
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What do you think?
Update memorization card.
COVID Year 8 limits, continuity, IVT.notebook
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Evaluate.
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What do you think?
COVID Year 8 limits, continuity, IVT.notebook
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What do you think?
COVID Year 8 limits, continuity, IVT.notebook
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What do you think?
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What do you think? What do you think?
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Evaluate the two limits
Evaluate the limit
Evaluate the limit
Evaluate the limit
Evaluate the limit
COVID Year 8 limits, continuity, IVT.notebook
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algebraically
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Hierarchy of DominanceGraph the following functions:
i(x) = x2f(x) = xxj(x) = ln(x)h(x) = exg(x) = x!
If all else fails....
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Example:
Given , is this function continuous at x = 1?
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Example:
Is this function continuous at x = 3?
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Example:Find the value of k that would make the following function continuous at x = 5.
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Example:Find the value of k that would make the following function continuous at x = 3.
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Example:Is the function continuous at x = 0?
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Example:For what values of x is there a discontinuity in the graph of:
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Example:Show that f(x) = is continuous everywhere.
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Continuity on an Interval
If a function, f(x), is continuous at each number in an open interval (a,b), then we say f(x) is __________________________________.
This also applies to infinite open intervals like (∞, a), etc.
If f(x) is continous on (∞, ∞) we say that f(x) is _________________
_____________________.
Example:
COVID Year 8 limits, continuity, IVT.notebook
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What about on a closed interval?
For a function to be continuous at an endpoint of an interval its value at the endpoint must be equal to the appropriate onesided limit at that endpoint.
In other words:
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Example:What can you say about the continuity of the function ?
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Continuity of CompositionsWhy even learn this?It will be useful for calculating limits of compositions of functions.
Theorem:If and if the function f is continuous at L, then
Theorem:
1. If the function g is continuous at c, and the function f is continuous at g(c), then the composition f(g(x)) is continuous at c.
2. If the function g is continuous everywhere and the function f is continuous everywhere, then the composition f(g(x)) is continuous everywhere.
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Example: What do you think?
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Pushing the IVT further...
Given [a, b], I would like you to plot the points
(a, f(a)) and (b, f(b)) such that:
1) Pick a height in which f(a)>0 and f(b) <0.
2) Draw a continuous function from f(a) to f(b).
3) We will compare it to mine.
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My graph is below.
Do we have the exact same graph?
What do you notice about the two? Is there anything they have in common?
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Memorization Card
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continuous
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Homework
Page 66
30)
33)
Page 76
1)
3)
4)
Page 97
2)
5)
Page 8586
14) At what values of x is f continuous.
47) Find a value for a so that the function
is continuous.
58) Which of the following statements about the function
is not true?
(A) f(1) does not exist
(B) exists
(C) exists
(D) exists
(E)
=2
=(1)(0)=0
=5/21
[1,0) U (0, 1) U (1, 2) U (2, 3)
x = 4/3
Choice A
1
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Homework
page 77 #69
page 9798 #10, 13, 40
10) Find the limit:
Find the limit
13) Find the limit:
40) What value should be assigned to k to make f a continuous function?
=2
= 0
= 0
k = 1/2
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