Section 4.5: Indeterminate Forms and? L

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Section 4.5 Indeterminate Forms and?
LHospitals Rule

• Practice HW from Stewart Textbook
• (not to hand in)
• p. 303 5-39 odd

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• In this section, we want to be able to calculate
  limits
• that give us indeterminate forms such as and
  .
• In Section 2.5, we learned techniques for
  evaluating
• these types of limit which we review in the
  following
• examples.

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• Example 1 Evaluate
• Solution

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• Example 2 Evaluate
• Solution

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• However, the techniques of Examples 1 and 2 do
  not
• work well if we evaluate a limit such as
• For limits of this type, LHopitals rule is
  useful.

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LHopitals Rule

• Let f and g be differentiable functions where
• near x a (except possible at x a). If
• produces the indeterminate forms , ,
  or , ,
• then
• provided the limit exists.

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• Note LHopitals rule, along as the required
• indeterminate form is produced, can be applied as
  
• many times as necessary to find the limit.

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• Example 3 Use LHopitals rule to evaluate
• Solution

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• Example 4 Use LHopitals rule to evaluate
• Solution

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• Example 5 Evaluate
• Solution

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• Note! We cannot apply LHopitals rule if the
  limit
• does not produce an indeterminant form ,
  , ,
• or .

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• Example 6 Evaluate
• Solution

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• Helpful Fact An expression of the form ,
  where
• , is infinite, that is, evaluates
  to or .

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• Example 7 Evaluate .
• Solution In typewritten notes

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Other Types of Indeterminant Forms

• Note For some functions where the limit does not
  
• initially appear to as an indeterminant , ,
  , or
• . It may be possible to use algebraic
  techniques to
• convert the function one of the indeterminants
  ,
• , , or before using LHopitals
  rule.

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Indeterminant Products

• Given the product of two functions , an
• indeterminant of the type or
  results
• (this is not necessarily zero!). To solve this
  problem,
• either write the product as or
  and evaluate
• the limit.

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• Example 8 Evaluate
• Solution

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• Example 9 Evaluate
• Solution In typewritten notes

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Indeterminate Differences

• Get an indeterminate of the form
  (this is not
• necessarily zero!). Usually, it is best to find a
  common
• factor or find a common denominator to convert it
  into
• a form where LHopitals rule can be used.

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• Example 10 Evaluate
• Solution

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Indeterminate Powers

• Result in indeterminate , , or
  . The natural
• logarithm is a useful too to write a limit of
  this type in
• a form that LHopitals rule can be used.

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• Example 11 Evaluate
• Solution (In typewritten notes)