1'5 Infinite limits

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1.5 Infinite limits

• "I never got a pass mark in math ... Just imagine
  -- mathematicians now use my prints to illustrate
  their books." -- M.C. Escher

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Objective

• To describe infinite limits

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Black holes

• Start with any number
• Count the number of even digits, the number of
  odd digits, the total number of digits.
• Write that 3-digit number
• Repeat
• Repeat
• Repeat

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Ways limits DNE

• Limit from the left is different than the limit
  from the right
• Function increases or decreases without bound
• Function oscillates

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Function increases or decreases without bound

• If both the left and the right side approach
  infinity then
• If both the left and the right side approach
  negative infinity then

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Discontinuities

• 2 types
• Removable
• Non-removable

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Def. of a vertical asymptote

• If f(x) approaches infinity or negative infinity
  as x approaches c from the right or left then the
  line x c is a v. a. of the graph

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V. A. theorem

• The functions f and g are continuous on an open
  interval. If f(c) does not equal zero, g(c) 0,
  and g(x) is not zero for all other x in the
  interval then
• has a v. a. at x c

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In other words

• Look for zeros in the denominator and then check
  the numerator to see if it is a hole or an
  asymptote

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Examples
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More examples
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Limits and V. A.

• Find
• What do you know about the function?

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Left and right limits
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Cont..

• Check from the left
• Check from the right
• The limit is

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Properties of limits

• 1. Sum or difference
• 2. Product

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More properties

• 3. Quotient
• These are also true for negative infinity

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Practice problems