2.2 Limits Involving Infinity

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2.2 Limits Involving Infinity

• Finite Limits as
• The symbol for infinity does not represent a real
  number.
• We use infinity to describe the behavior of a
  function when the values in its domain or range
  outgrow all finite bounds.
• When we say the limit of f as x approaches
  infinity we mean the limit of f as x moves
  increasingly far to the right of the number line.
• When we say the limit of f as x approaches
  negative infinity, we mean the limit of f as x
  moves increasingly far to the left.

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• Looking at f(x) 1/x, we observe
• We can say that the line y 0 is a horizontal
  asymptote of the graph of f.

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Horizontal Asymptote
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Looking for Horizontal Asymptotes

• Use graphs and tables to find
• and identify all horizontal asymptotes of
• The horizontal asymptotes are y 1 and y -1.

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Finding a Limit as x Approaches Infinity

• Find
• The numerator is decreasing to a very small
  number. The denominator is increasing to a very
  large number. This causes the function to
  approach 0.

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Using Theorem 5

• Find

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Finding Vertical Asymptotes

• Find the vertical asymptotes of f(x) 1 / x².
  Describe the behavior to the left and right of
  each vertical asymptote.
• The values of the function approach infinity on
  either side of x 0.
• The line x 0 is the only vertical asymptote.

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Finding Vertical Asymptotes

• The graph of f(x) tan x (sin x) / (cos x) has
  infinitely many vertical asymptotes, one at each
  point where the cosine is zero.

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Modeling Functions for x Large

• Let
• and . Show that while f and g are quite
  different for numerically small values of x, they
  are virtually identical for x large.

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Finding End Behavior Models

• Find an end behavior model for
• a.
• b.

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Finding End Behavior Models

• Let f(x) x e-x . Show that g(x) x is a
  right end behavior model for f while h(x) e-x
  is a left end behavior model for f.
• On the right,
• On the left,

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Using Substitution

• Find