Limits

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Limits

• Why limits?
• What are limits?
• Types of Limits
• Where Limits Fail to Exist
• Limits Numerically and Graphically
• Properties of Limits
• Limits Algebraically
• Trigonometric Limits
• Average and Instantaneous Rates of Change
• Sandwich Theorem
• Formal Definition of a Limit

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Why limits?

• Limits help us answer the big question of how
  fast an object is moving at an instant of time.
  For Newton and Leibniz, this had to do with the
  velocity a planet moved in its orbit around the
  sun.

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Why limits?

• We might be more interested in the velocity of
  other things

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Why limits?

• The fundamental concepts of calculus - the
  derivative and the integral are both defined in
  terms of limits. We will see more of these as we
  learn how to use limits.

Derivative
Integral
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Why limits?

• So limits are like the engine under the hood of a
  car. We are mainly interested in driving the car
  and wont spend a lot of time thinking about what
  is happening under the hood, but we should have a
  basic understanding of how the engine works.

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What are limits?

• Limits describe the behavior of functions around
  specific values of x. They also describe the end
  behavior of functions.
• More specifically, limits describe where the
  y-value of a function appears to be heading as x
  gets closer and closer to a particular value or
  as x approaches positive/negative infinity.
• Lets look at these ideas a little closer.

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What are limits?

• Some important notes about limits
• a. Limits are real numbers, but we sometimes
  use to indicate the direction a function
  is heading.

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What are limits?

• Some important notes about limits
• b. Limits do not depend on the value of the
  function at a specific x value, but on where
  the function appears to be heading.

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What are limits?

• For a limit to exist, the function must be
  heading for the same y-value whether the given
  x-value is approached from the left or from the
  right, i.e. one-sided limits must agree.

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Types of Limits

• There are three basic forms of limits (
  )
• a. Limits at a finite value of x
• b. Infinite limits (vertical asymptotes)
• c. Limits at Infinity (horizontal
  asymptotes or end behavior)

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Infinite Limits

• Infinite limits occur in the vicinity of vertical
  asymptotes. Functions may approach positive or
  negative infinity on either side of a vertical
  asymptote. Remember to check both sides
  carefully.
• Also, remember to simplify rational expressions
  before identifying vertical asymptotes.

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Infinite Limits Ex. 1
Determine the limit of each function as x
approaches 1 from the left and from the right.
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Infinite Limits Ex. 2
Identify all vertical asymptotes of the graph of
each function.
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Limits at Infinity
Remember
logarithmic
polynomial
exponential
factorial
??????????
Evaluate
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Limits at Infinity
Video
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Limits at Infinity
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Limits at Infinity
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Where Limits Fail to Exist

• There are three places where limits do not exist

Jump Discontinuities
Vertical Asymptotes
Oscillating Discontinuities
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Limits Numerically 1
Use the TblSet (with Independent set to ASK) and
TABLE functions on your graphing calculator to
estimate the limit.
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Limits Numerically 2
Use the TblSet (with Independent set to ASK) and
TABLE functions on your graphing calculator to
estimate the limit.
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Limits Numerically 3
Use the TblSet (with Independent set to ASK) and
TABLE functions on your graphing calculator to
estimate the limit.
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Limits Graphically 1
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Limits Graphically 2
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Limits Graphically 3
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Limits Graphically 4
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Properties of Limits 1
Some examples
Thinking graphically may help here.
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Properties of Limits 1 Ex. 1
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Properties of Limits 1 Ex. 2
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Properties of Limits 1 Ex. 3
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Properties of Limits 2
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Properties of Limits 2 Ex. 1
Use the information provided here to evaluate
limits a d here
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Properties of Limits 2 Ex. 2
Use the information provided here to evaluate
limits a d here
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Properties of Limits 3
For example, evaluate the limit
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Properties of Limits 4
For example
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Properties of Limits 5
For example, evaluate the limit
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Properties of Limits 6
For example, given
Evaluate the limit
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Limits Algebraically

• In addition to Direct Substitution, there are
  many strategies for evaluating limits
  algebraically. In particular, we will focus on
  three of them
• Factor and Cancel
• Simplifying Fractions
• Rationalization

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Factor and Cancel
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Simplifying Fractions
Basic Strategy Multiply numerator and
denominator by 3(3x) and then simplify. You
could also find a common denominator for both
fractions in the numerator and then simplify that
first.
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Rationalization
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Trigonometric Limits
There are two special trigonometric limits
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Trigonometric Limit Ex. 1
Basic strategy
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Trigonometric Limit Ex. 2
Strategy Multiply numerator/denominator by 4
Let and note that as
,