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Limits and Continuity
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Table of ContentsLimitsLimits of End BehaviorThe Indeterminable Form of 0/0Trig LimitsContinuityDifference Quotient
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A rubber ball is dropped and bounces back up to half the height it was on the previous bounce. Given this scenario, does the ball ever come to rest? What is the approximate height as time goes to ∞?
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A limit allows a function, f(x), to have a value as the function approaches x.
In the previous example, the balls height was zero as time went to ∞, even if the object never stopped moving.
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Consider the functions as x approaches 3.
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As the function f(x) approaches x=3 what is the value f(x)?
As the function g(x) approaches x=3 what is the value g(x)?
A limit describes what happens to the function as it gets closer and closer to a certain value of x. The function doesn't have to have a value at that x for the limit to exist.
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We can find a limit by substituting into the function the value of x.
So in the previous example f(x)= x-3, f(3)=0, which the graph shows.
What about ?
Substitution doesn't work. But since a limit is defined by getting close to that value we can look at the graph and see that the function has a value approaching -3 as x approaches 0 from both the left and right. Algebraically, we can factor and reduce and we also will get a limit of 0.
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LimitsLimits describe what happens to a function as x approaches a value.
is read "The limit of f of x, as x approaches c, is L.
For the previous example of f(x)= x-3:
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Limits of a Composite Function can be found by finding the limits of the individual terms.
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Ex:
Approaches 1 from the right only.
Approaches 1 from the left only.
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1 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
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2 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
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6 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
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Use graphing calculator to determine:
Using the second function, find the value of x=2.What does this demonstrate?
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Using a graphing calculator with
Go to 2nd -> TBLSET TblSTART= -2 Auto #Tbl=.5 Auto
Now use 2nd -> TABLE What do you observe? What happens at x = 2?
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8 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
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9 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
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10 Find the indicated limit, if it exists. If it doesn't exist enter 0.0
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When a limit goes to infinity we are looking for any horizontal asymptotes and a number of things can happen.
The first example isn't a problem, as was seen in the last section.
But what about the second and third examples?
Not all # are the same.
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Consider is the limit 1?
Reduce the rational before substituting in # . The limit is # .
In calculus, divide each term by the highest power of x, then take the limit.
Remember that #/# is 0.
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Be aware that some of the limits are going to -# .
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Short cut: Compare the highest power in the numerator and denominator.
Same power, divide the coefficents
If numerator is greater, #
If the denominator is greater, 0
What is the highest power in the numerator and denominator?
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by the short cut, the highest powers are the same. But be careful before dividing the coefficents?
reduces to 2. But the limit is going to -# . Since the denominator is always positive, only the numerator is negative.
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11
A
B
C
D
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12
A
B
C
D
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13
A
B
C
D
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Why? Which is getting bigger faster?
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Use conjugates to rewrite the expression as a fraction and then solve like # /# .
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Try it.
Ans: 5/2
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Limits that go to # /# or # -# are called indeterminate forms. The third type we study are 0/0.
Think back to the example
When we graphed the function and looked at the table of values the limit was 4, but subbing in 2 was 0/0.
So what gives?
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Think back to Algebra, when you plugged a number into an equation and got zero we said that the number was a root. Since we are getting 0/0, that means the numerator and denominator share a root. So this means we should be able to factor and reduce.
Remember a limit is what the function is doing as it approaches a value from the left and right. x# 2 but the function can approach it.
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Try:
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Try: Find:
a DNE
b -5
c 4
answers
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14 Find the limit of
A
B 4
C 0
D Does not exist
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15 Find the limit of
A
B 4
C 0
D Does not exist
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16 Find the limit of
A
B -1
C 0
D Does not exist
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17 Find the limit of
A
B -1/2
C 0
D Does not exist
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18
A
B
C
D Does not exist
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Review of Limits1) plug in a for x, if it equals a number, there's your limit.
2) If when a is plugged gives:
A) :Factor and reduce. Try plugging in again.
B) i) Divide each term by the highest exponent term ii)Short cut: compare highest exponent of the numerator with the highest exponent of the denominator. a) numerator greater -># b) denominator greater -> 0 c) degrees are the same->lead coefficient of numerator lead coefficient of denominator (Note: watch your signs when using -# ) C) :Multiply by conjugate and then try plugging in again D) : This is # but check to see that
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If you can plug in and get a number, good. But what if you get an indeterminable form?
3 Trig Limits to Know
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19 Find the limit of
(Answer in fraction form if not an integer.)
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20 Find the limit of
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21 Find the limit of
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22 Find the limit of
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23 Find the limit of
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a b c de f g h
At what points do you think the graph is continuous?At what points do you think the graph is discontinuous?
What should the definition of continuous be?
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AP Calculus Definition of Continuous
1) f(a) exis ts
2) exis ts
3)
This definition shows continuity at a point on the interior of a function.
For a function to be continuous, every point in its domain must be continuous.
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Continuity at an EndpointReplace step 3 in the previous definition with:
Left Endpoint:
Right Endpoint
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Types of Discontinuity
Infinite Jump Removable Essential
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24 Given the function decided if it is continuous or not. If it is not state the reason it is not.
A continuousB
C
D
f(a) does not exis t for all a
does not exis t for all a
is not true for all a
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25 Given the function decided if it is continuous or not. If it is not state the reason it is not.
A continuousB
C
D
f(a) does not exis t for all a
does not exis t for all a
is not true for all a
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26 Given the function decided if it is continuous or not. If it is not state the reason it is not.
A continuousB
C
D
f(a) does not exis t for all a
does not exis t for all a
is not true for all a
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27 Given the function decided if it is continuous or not. If it is not state the reason it is not.
A continuousB
C
D
f(a) does not exis t for all a
does not exis t for all a
is not true for all a
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Removable discontinuities come from rational functions.
A piecewise function can be used to fill the "hole".
What should k be so that g(x) is continuous?
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28What value(s) would remove the discontinuity(s) of the given function?
A -3
B -2
C -1
D -1/2
E 0
F 1/2
G 1
H 2
I 3
J DNE
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29What value(s) would remove the discontinuity(s) of the given function?
A -3
B -2
C -1
D -1/2
E 0
F 1/2
G 1
H 2
I 3
J DNE
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30What value(s) would remove the discontinuity(s) of the given function?
A -3
B -2
C -1
D -1/2
E 0
F 1/2
G 1
H 2
I 3
J DNE
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31What value(s) would remove the discontinuity(s) of the given function?
A -3
B -2
C -1
D -1/2
E 0
F 1/2
G 1
H 2
I 3
J DNE
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Making a Function Continuous
, find a so that f(x) is continuous .
Both 'halves' of the function are continuous. The concern is making
Solution:
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32What value of k will make the function continuous?
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33What value of k will make the function continuous?
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Characteristics of a function on closed continuous interval.
1) Somewhere on the interval there will be a maximum and a minimum.
i ii iii iv
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When discussing Maximum and Minimum, there is a difference between Absolute Maximum and Relative Maximum.
Think of dropping a ball.The starting point is absolute max.The starting point and each bounce are relative max.
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Characteristics of a function on closed continuous interval.
*2) Intermediate Value Theorem- if f(x) is a closed continuous interval on [a,b], then f(x) takes on every value between f(a) and f(b).
* IVT is a theorem that can be used as a justification on the AP exam.
af(a)
b
f(b)
This comes in handy when looking for zeros.
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Can you use the Intermediate Value Theorem to find the zeros of function?
X Y
-2 -6
-1 -2
0 3
1 5
2 -1
3 -4
4 -2
5 2
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Can you use the Intermediate Value Theorem to find the zeros of function?
X Y
-3 4
-2 1
-1 -1
0 1
1 2
2 3
3 -2
4 2
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34Give the letter that lies in the same interval as a zero of this continuous function.
A B C DX 1 2 3 4 5
Y -2 -1 0.5 2 3
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35Give the letter that lies in the same interval as a zero of this continuous function.
A B C DX 1 2 3 4 5
Y 4 3 2 1 -1
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36Give the letter that lies in the same interval as a zero of this continuous function.
A B C DX 1 2 3 4 5
Y -2 -1 -0.5 2 3
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Draw a possible graph of traveling 100 miles in 2 hours.
Distance
Time
100
t
d
2
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Using the graph on the previous slide: What is the average rate of change for the trip?
Is this constant for the entire trip?
What formula could be used to find the average rate of change between 45 minutes and 1 hour?
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The slope formula of represents the Velocity or
Average Rate of Change. This is the slope of the secant line from (t1,d1) to (t2,d2).
Suppose we were looking for Instantaneous Velocity at 45 minutes, what values of (t1,d1) and (t2,d2) should be used?
Is there a better approximation?
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The Difference Quotient gives the instantaneous velocity, which is the slope of tangent line at a point.
A Derivative is used to find the slope of a tangent line. So Difference Quotient can be used find a derivative algebraically.
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Example of the Difference QuotientFind the slope of the tangent line to the functionat x=3.
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Example of the Difference QuotientFind an equation that can be used to find the slope of the tangent line at any point on the function
To find the slope at x=2, sub 2 into 9x2. So the slope at x=2 is 36.
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40What expression can used to find the slope of any tangent line to
A
B
C
D
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Now try finding the slope of a tangent of a graph.
y1=sin(x)Find the slope at x=# / 4
Try ZOOM IN at the point till it looks like a line in the viewing window. (To get calculator to zoom at x=# /4, use 2nd CALC 1:Value, then zoom once the cursor is on π/4.)
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The cursor is on the function at x=# /4 . We don't want round on the AP exam until the end but who wants to copy down all those numbers?
2nd QUIT will take you to the computation screen with x and y still equal to the point on the graph. STORE x->A and y->B using the STORE key and ALPHA.
Go back to graph and "bump" the cursor over to a point close by.2nd QUIT again and this time STORE x->C and y->D.
Calculate the slope (D-B)/(C-A)= .707 (now you can round)
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The calculator does have a built in derivative key for graphs.
It is dy/dx. When the cursor is where you want it, use 2nd CALC 6: dy/dx (You may need to zoom in a few times to take the derivative where you want it)
These both stand for the derivative of a function.
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