Limits in Calculus JON PIPERATO CALCULUS GRADES 11-12
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Limits in Calculus, an Introduction Limits are a major
component to Calculus, which is used everyday Integrals take the
limit and use it to find an area under a curve Finding surface
area, volume, distances etc. Electric charge All engineering and
all science
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Navigation Through the Unit All buttons such as these will take
you to the next or previous slide Any information button will take
you to the beginning of the lesson that is specified The home
button will bring you back to the main menu The question mark will
take you to the review problems from that lesson Reveals answers to
example problems
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Main Menu Lesson 1: Intuitive Definition and Graphing Lesson 2:
Limits in Tables Lesson 3: Limits that fail to exist Lesson 4:
Algebraic Limits Lesson 5: Piecewise Functions Lesson 6: Infinite
Limits Lesson 7: Discontinuity **Clicking the information button
will take you to the beginning of each lesson **? Button takes you
to review problems
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Lesson 1: Intuitive Definition & Graphing A limit of a
function is described as the behavior of that function as it
approaches a specific point EX: This reads: as x approaches c, the
function f(x) approaches the real number L. In other words, as x
gets closer to c, that f(x) gets closer to LTHIS DOES NOT MEAN
f(c)=L,
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Limits in Graphing lim f(x)= 2 X 4 lim f(x)= 4 x-1 lim f(x)= -2
X -4 + lim f(x)= 3 X -4 - lim f(x)= DNE X -4 lim f(x)= -1 X 2 - lim
f(x)= -1 X 2 + f(2)= 3
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For limits approaching an x- value with two points, if a
direction is not specified, the limit DNE Ex. lim f(x)= DNE X -4 If
a direction is specified, use you finger and follow the line from
that direction until you reach the point. Ex. lim f(x)= -2 X -4 +
If a specific function is given, you are looking for the point that
is not connect to the lines in the graph Ex. f(2) = 3 Limits in
Graphing Continued
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Examples of Intuitive Definition Write the following limits in
sentence form: 1) 2) 3) Click here to see answers!
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Answers to Intuitive Definition 1)the limit of f(x) as x
approaches a is L 2) the limit of 1/x as x approaches infinity is 0
3)the limit as 1/x as x approaches 0 is infinity
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Limit Graphing Example lim f(x)= X -4 lim f(x)= X -2 lim f(x)=
X 1 lim f(x)= X 1 - lim f(x)= X 1 + f(1)= f(-2)= Click here to see
the answers!
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Graphing Limits Answers lim f(x)= X -4 lim f(x)= X -2 lim f(x)=
X 1 lim f(x)= X 1 - lim f(x)= X 1 + f(1)= f(-2)= -2 3 DNE -3 4 2
5
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You Have Completed Lesson 1! Summary Limits are defined as the
characteristics of the function as it approaches a specific point.
Intuitive Definition is writing a limit in sentence form Graphing
limits can be used to find a specific location of functions as they
approach a specific point on a graph
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Lesson 2: Limits in Tables There are multiple ways to solve for
limits in Calculus This is a second option to solving limits For
this lesson, you will need a TI-84 calculator to complete the
tasks
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Limits in Tables The first step of solving limits through
tables is to identify what variable x is going to in the limit lim
f(x 2 -9/x+3)= x -3 Next, knowing that x is approaching -3, you
must construct your table using numbers close to the variable in
the statement above. Numbers such as -3.1, 3.01, 3.001 & -2.9,
-2.99, -2.999 Then put the equation into the calculator Finally use
the calculator to find the missing values and estimate what the
pattern of the table is Dont panic, we will go through problems
together
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Limits in Tables Here is an example of what the table from the
previous problem would look like -3.1-6.1 -3.01-6.01 -3.001-6.001
-3 -2.999-5.999 -2.99-5.99 -2.9-5.9 Once the numbers are found, you
must use the pattern to figure out was - 3 is equal toit is equal
to -6 Therefore lim f(x)= -6 x-3
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Limit in Tables Examples Try these limits with a partner: Click
here to see the answers!
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Limits in Tables Answers = 40 = 1/2 = 6
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You Have Completed Lesson 2 Summary In conclusion we discovered
a way to solve for a limit using our calculators and tables Without
the proper calculator this problem will be a pain, so please see me
if you must borrow one!
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Lesson 3: Limits That Fail to Exist A limit doesn't exist if
the function is not continuous at that point. To check if a limit
exists or not, graphically, you must approach it from the left and
right side and if they are not equal, they do not exist. One type
of limit that fails to exist is a jump. A jump can be found in the
graphing of the function |x|/ x Another is a vertical asymptote A
vertical asymptote is when x=0 Lastly, f(x) oscillates between two
fixed values as x approaches c
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Examples Of Discontinuities Match the following discontinuities
with the possible graph |x|/ x 1/x 2 Sin (1/x) Click here to see
the answers!
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Answer to Discontinuities |x|/ x Sin (1/x) 1/x 2
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You Have Completed the Lesson!
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Lesson 4: Algebraic Limits We can solve certain limits with our
knowledge of algebra! All we have to do is plug the number x is
going towards in the equation But of course there is a catch! You
cannot just simply plug that number in if it makes the equation
false. We cannot make the denominator zero
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Algebraic Limits An example of the information on the previous
slide is: Since plugging -2 in will give us zero in the
denominator, we must do some solvingwith algebra!!!! However, if it
plugs in without a problem, then you just solve!
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Algebraic Limits There are three ways to solve for a limit 1)
Limits by direct substitution 2) Limits by FactoringYay! 3) Limits
by Rationalization A video on the next slide will explain how to
solve the following limits
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Algebraic Limits Video
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Algebraic Limits Limits by direct substitution Exactly how it
sounds Just plug the number in! Limits through factoring Use
factoring to cross out unwanted denominator Limits through
Rationalization Use the reciprocal to get rid of the unwanted
denominator
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Algebraic Limit Example: Select one to be your answer: A) 10 B)
5C) 2D) 2 A) 10 B) 5C) 2D) 2
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You Are Incorrect, Please try again Refer to previous slide if
necessary, or raise your hand for assistance
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You are correct! Please return to menu!
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Algebraic Limits Get with a partner, and solve the following
limits algebraically through factoring 1) 2) 3) The button for the
answers will be on the next page
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Algebraic Limits Stay with your partner and work on the
following rationalization problems 1) 2) Click here to see the
answers!
You Have Finished Lesson 4: Algebraic Limits Summary In summary
there are three different ways to solve for an algebraic limit
Direct substitution Factoring Rationalization Watch the video
before coming to class to get a better understanding of the
lesson!
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Lesson 5 Piecewise Functions A piecewise function look like
this : We already have the skills from previous sections to solve
this problem so do not let looks deceive you! It is called
piecewise because it is broken into pieces on a graph, but it is no
different then solving ordinary functions!
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Limits of a Piecewise Function Example: Solve for f(5) Use the
function that satisfies the number five Clearly, you must use the
second since 5>0 You then just you substitution from last
section (5) 2 = 25
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Limits of a Piecewise Function Lets try one with a limit lim
f(x) = x -2 - So first we must determine which piece to use Then
use direct substitution We can determine that we must use the first
piece since x is less than negative 2 (coming from the left) Using
direct substitution we plug in -22(-2)+8= -4+8= 4 Therefore lim
f(x) = x-2 - 4
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Limits of a Piecewise Function Review Time for you to try some
on your own! lim f(x) = x5 Will the answer be A) 10 A) 10 B)5 B)5
C) 0 C) 0
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You Are Incorrect, Please try again Refer to previous slide if
necessary, or raise your hand for assistance
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You are correct! Please Proceed!
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One more example just to be sure! lim f(x) = x0 What is your
answer? A) 4 A) 4 B) 8 B) 8 C) 5 C) 5 Limits of a Piecewise
Function
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You Are Incorrect, Please try again Refer to previous slide if
necessary, or raise your hand for assistance
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You are correct! Please return to main menu!
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You Have Finished Lesson 5: Piecewise Functions Summary Solving
for a piecewise function is easy for us, because we already learned
the process! The only difference is that the function we are
looking at is broken up into different parts We will look at why
they are broken up in an upcoming lesson Any questions?
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Lesson 6 Infinite Limits These problems can be extremely
simpleas long as you learn the three rules!!!!! The first rule: If
the powers are the same (of the variable) in the denominator and
numerator, then your answer will be the coefficients of the
variables. The second rule: If the power of the variable in the
numerator is higher then the denominator, then your answer is
(infinite) The third rule: If the power of the variable in the
numerator is less than the denominator, your answer is 0
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Infinite Limits Refer to this table if you get confused
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Infinite Limits Lets try some examples! First lets find the
variables and what power they are to Since they are to the same
power, we must you rule number one, which is to take their
coefficient. So the coefficient of x is one and the coefficient of
3x is 3. Therefore your answer would be 1/3
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Infinite Limits Example: Looking at the variables, we can
conclude that they have different powers, and the denominator is
bigger. Therefore we must use rule number three. With our knowledge
we know that rule three makes our answer 0
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Infinite Limits One last example! We can conclude that the
numerators power is greater than the denominator, which means we
will use the second rule. But wait!!!!! do not ignore the negative
sign in front of the infinity symbol In this case the negatives
will cancel out giving you a positive
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Infinite Limits Examples Try some of these examples! 1) 2) 3)
4)
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Infinite Limit Answers = 1/2 = 0 = - = 1/3
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You Have Completed Lesson 6!
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Lesson 7 Asymptote Graphs (Discontinuities) Remember in lesson
5, piecewise functions, I said we would look at why they were in
pieces? There are three different reasons for asymptotes in a graph
that will be touched on in the lesson JumpsPiecewise functions A
limit at infinity (1/x or 1/x 2 ) A removable discontinuity
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Discontinuities Jump
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Discontinuities Limit at infinity
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Discontinuities Removable
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Lesson 7 Discontinuity We talked about the three types of
discontinuities and examples of each Removable Jumps Limits at
infinity Which of the three types are scene in the graphing
approach of solving limits? A) RemovableRemovable B)JumpsJumps C)
Limits at InfinityLimits at Infinity
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You Are Incorrect, Please try again
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You are correct! Please to the main menu!
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The Lesson is Complete! Click the button to return to the title
slide for the next student!