EVALUATING LIMITS ANALYTICALLY

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EVALUATING LIMITS ANALYTICALLY

• Section 1.3

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When you are done with your homework, you should
be able to

• Evaluate a Limit Using Properties of Limits
• Develop and Use a Strategy for Finding Limits
• Evaluate a Limit Using Dividing Out and
  Rationalizing Techniques
• Evaluate a Limit Using the Squeeze Theorem

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SOME BASIC LIMITS

• Let and b and c be real numbers and let n be a
  positive integer.

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Evaluate

• 5
• -3
• Does not exist

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Evaluate

• 1
• -1
• 5
• Does not exist

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Properties of Limits

• Let and b and c be real numbers, let n be a
  positive integer, and let f and g be functions
  with the following limits
• 1. Scalar multiple
• Sum or difference
• Product
• Quotient
• 5. Power

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Evaluate

• -4
• -2
• 1
• Does not exist

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Limits of Polynomial and Rational Functions

• If p is a polynomial function and c is a real
  number, then .
• If r is a rational function given by
  
• and c is a real number
• such that then

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Evaluate

• 5
• 0
• -5
• Does not exist

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Evaluate the function at x 2

• 1
• DNE

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The Limit of a Function Involving a Radical

• Let n be a positive integer. The following limit
  is valid for all c if is n odd, and is valid for
  if n is even.

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The Limit of a Composite Function

• Let f and g be functions with the following
  limits
• Then

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Limits of Trigonometric Functions

• Let c be a real number in the domain of the given
  trigonometric function.

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STRATEGIES FOR FINDING LIMITS

• Functions That Agree at All But One Point
• Let c be a real number and let for
  all x in an open interval containing c. If the
  limit of g as x approaches c exists, then the
  limit of f also exists and
  .

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A Strategy for Finding Limits

• Learn to recognize which limits can be evaluated
  by direct substitution.
• If the limit of f(x) as x approaches c cannot be
  evaluated by direct substitution, try to find a
  function g that agrees with f for all x other
  than x c.
• Apply
• .
• Use a graph or table to reinforce your
• conclusion.

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Dividing Out Techniques

• Example

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Evaluate.

• 0
• Does not exist
• 1/16

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Rationalizing Techniques

• Example

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Evaluate the exact limit.

• .25
• 0.0

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TWO SPECIAL TRIGONOMETRIC LIMITS
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Evaluate the exact answer.

• 5.0
• 0.0