Horizontal and Vertical Asymptotes

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• Horizontal and Vertical Asymptotes

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Vertical Asymptote

• A term which results in zero in the denominator
  causes a vertical asymptote when the function is
  graphed, providing that the function is in its
  lowest terms. The vertical asymptote is found at
  the term which causes the zero in the denominator.

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Example 1

• Find the vertical asymptote in the following
  function

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• Note that x 5 results in a zero in the
  denominator. The fraction is in simplest form.
  Therefore, there is a vertical asymptote at x 5

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Example 2
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• would appear to have a
  vertical
• asymptote at x -1. The reason that it does not
  is that the fraction may be re written as
•  

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• This simplifies to (x-1), which does not result
  in a zero in the denominator. This does not
  result in a vertical asymptote.

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Horizontal Asymptotes
Horizontal asymptotes, when they exist, are
determined by the value approached by the
function as x gets either extremely large or
extremely small. When graphed, asymptotes are
expressed as dashed lines which the values
graphed approach but never meet
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Three types of rational expressions

• There are three types of rational expressions, as
  determined by the relationships of the greatest
  power in the numerator and denominator.
• If the greatest power in the denominator is
  greater than the greatest power in the numerator,
  there is a horizontal asymptote at y 0

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Example 3

• The greatest
  power in the numerator is 1. The greatest power
  in the denominator is 2. Therefore, as x gets
  larger and larger, there is a horizontal
  asymptote at x 0
  

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• If the greatest power in the numerator is equal
  to the greatest power in the denominator, there
  is a horizontal asymptote at the ratio of the
  leading coefficients.

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Example 4

• The greatest power in the numerator is one. The
  greatest power in the denominator is one. The
  ratio of the leading coefficients is 3/1, or 3.
  There is a horizontal asymptote at 3.

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Proof using a table of values

•  

X F(x)
-1000 3.025
-100 3.269
-8 28
-7.001 25003
-6.999 -25000
-6 -22
1000 2.975
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Proof using Algebra Multiply numerator and
denominator by 1/x

• Note that the numerator simplifies to 3 4/x,
  and the denominator simplifies to 1 7/x
• At extremely high and extremely low values of x,
  the values of -4/x and 7/x approach zero,
  resulting in a horizontal asymptote at 3/1, or 3

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• If the greatest power in the numerator is one
  greater than the greatest power in the
  denominator, an oblique asymptote generally
  results.