Limits Review by Sara Barton

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Limits Reviewby Sara Barton

• A Limit is a value that a function approaches as
  x approaches some fixed value

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A limit is the y value a function is trying to
reach as it approaches some number I t may
or may not reach the number
Understanding Limits Verbal
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Understanding LimitsNumerical
Assume that this is the table corresponding with
a graph in Y

• x y
• .1 .268
• .2 .253
• .3 ERR
• .4 .247
• .5 .236

Limit as x .2 is .253 We can estimate the
limit as x .3 Using the values shown
It looks like .25
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Understanding LimitsAnalytical
Indeterminate More work to do
Whats next?
FACTOR You try!!
Since by plugging in 2, the value was
indeterminate, there is point discontinuity at x2
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Understanding LimitsAnalytical
Undefined when 0 is the denominator Therefore
the limit does not exist in this case
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Understanding LimitsGraphical

• Limits DO NOT EXIST
• at a jump
• At a vertical asymptote

The limit exists when we can see that the right
and left side of the function are heading to the
same place
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Understanding Limits Graphical
Do the limits Exist?
YES! Both the left and right functions are going
to the same place YES! There is a corresponding
value on the function at x0 YES! Both the left
and right functions are going to the same place
a
b
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Limits Involving Infinity

• Limits as x approaches infinity act like
  horizontal asymptotes

Examples
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Limits Involving InfinityShortcuts!!!
Use the leading exponents to find the limit
Top Heavy
DOES NOT EXIST
Bottom Heavy
Limit is 0
Same exponent
Limit is ratio of leading coefficients The limit
is 3
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LHopitals Rule

• Is used to evaluate limits when the function is
  indeterminate when solving

(When a is a finite number or positive or
negative infinity)
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LHopitals RuleHelpful Hints

• If it is indeterminate but not in quotient form,
  put the function in quotient form
• Take the derivative of the numerator and
  denominator separately, DO NOT use the quotient
  rule
• Check to see if the result is indeterminate after
  using LHopitals rule
• If the result is indeterminate, take the second
  derivative of the original function
• You can continue to derive if the result
  continues to be indeterminate

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Now You try!!Practice Problems

• Compute

• 0
• 11/4
• 1
• 3
• the limit does not exist

http//www.calculus.org/ Calculus Problems with
Step By Step Solutions
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Step By Step
Plug the limit, 2, into the equation
Therefore the solution is indeterminate this
means there is more work to do!!
factor both the numerator and denominator
Divide out (x-2) and solve again
Plug the limit, 2, into the equation again
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Step by steptry LHopital!!
We know this is indeterminate
Sooo take the derivative of the top and bottom
Plug in the limit to the new function
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Practice Problem 2

• Graph

Using the graph, determine
Amscos AP Calculus Maxine Lifshitz
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Step By step
Since the functions involve inequalities we must
include these on the y screen in order to view
the correct graph
You can find the inequality signs by pressing the
2nd Math keys
The graph looks like this
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Step By Step
a) To find the limit as x approaches -1, find

• b) Finding the limit as x approaches 0 will be
  more complicated since there are two different
  functions to the left and right of the y axis.
  Therefore we must find the left and right limits
  of f (x)

Right limit ()
Left limit (-)
Therefore the limit does not exist
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Practice Problem 3

• Consider the function

Determine the values of constants a and b so that
exists
http//www.calculus.org/ Calculus Problems with
Step by Step solutions
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Step by step

• Begin by computing one-sided limits at x2 and
  setting each equal to 3.

Right limit
Left limit
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Now solve the system of equations
a2b 3 and b-4a 3
a 3-2b so that b-4(3-2b) 3 if b-12 8b 3
If 9b 15
.
So
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More Practice

• Find the value of

Answer
d) 2
c) 1 d) 2 e) 8

• 0
• ½

Amscos AP Calculus Maxine Lifshitz
c) 1
Answer

• 0
• 1/7

c) 1 d) 7 e) indeterminate
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More Practice
Sketch the graph of a function f(x) that has all
of the following properties

• Has a hole at x -1
• Continuous for all x 3, -1
• Lim f(x) - 8
• Lim f(x)0
• Lim f(x) -8

X 3
X 8
X -8
Amscos AP Calculus Maxine Lifshitz
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Answer

• Actual graph may vary.
• The sections in red are necessary.

y
x
Horizontal asymptote
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Now You Know Limits!!

• BUT
• If you still need help check out

http//www.calculus-help.com/funstuff/phobe.html
For more online lessons on
Lesson 1 What is a Limit? Lesson 2 When Does a
Limit Exist? Lesson 3 How do you evaluate
limits? Lesson 4 Limits and Infinity Lesson 5
Continuity Lesson 6 The Intermediate Value
Theorem