Infinite Limits and Limits to Infinity: Horizontal and Vertical Asymptotes

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Infinite Limits and Limits to Infinity
Horizontal and Vertical Asymptotes
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Recall

• The notation tells us how the limit
  fails to exist by denoting the unbounded behavior
  of f(x) as x approaches c.
• Infinity is not a number!

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Properties of Infinite Limits

• Let c and L be real numbers and let f and g be
  functions such that
• and
• Sum or difference
• Consider

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Properties of Infinite Limits

• Let c and L be real numbers and let f and g be
  functions such that
• and
• Product if L gt 0
• if L lt 0
• Consider

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Properties of Infinite Limits

• Let c and L be real numbers and let f and g be
  functions such that
• and
• Quotient
• Consider

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Definition - Vertical Asymptotes

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

vertical
asymptote
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Determining Infinite Limits
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The pattern
and c is a positive integer
Is c even or odd? Sign of p(x) when x c
odd positive
odd negative
even positive
even negative
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Using the pattern
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Using the pattern
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Limits at Infinity

• denotes that as x
  increases without bound, the function value
  approaches L
• L can have a numerical value, or the limit can be
  infinite if f(x) increases (decreases) without
  bound as x increases without bound

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Horizontal Asymptotes

• The line y L is a horizontal asymptote of f if
• or
• Notice that a function can have at most two
  HORIZONTAL asymptotes (Why?)

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0
0
Horizontal Asymptote(s)__________
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2
2
Note It IS possible for a graph to cross its
horizontal asymptote!!!!!!
Horizontal Asymptote(s)__________
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1
0
Horizontal Asymptote(s)__________
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0
0
Horizontal Asymptote(s)__________
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Theorem Limits at Infinity

1. If r is a positive rational number and c is any
   real number, thenThe second limit is valid
   only if xr is defined when x lt 0

0
0
0
0
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Using the Theorem
0
0
2
0
0
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Guidelines for Finding Limits at 8 of Rational
Functions
less than

1. If the degree of the numerator is ___________ the
   degree of the denominator, then the limit of the
   rational function is ___.
2. If the degree of the numerator is _______ the
   degree of the denominator, then the limit of the
   rational function is the __________________
   _______________________.
3. If the degree of the numerator is ___________ the
   degree of the denominator, then the limit of the
   rational function _______________.

0
equal to
the ratio of the
leading coefficients
greater than
is infinite
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Using the Guidelines
0
2
1 3
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