2'6 Limits Involving Infinity

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2.6 Limits Involving Infinity
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Definition
The notation
means that the values of f (x) can be made
arbitrarily large (as large as we please) by
taking x sufficiently close to a (on either side)
but not equal to a.
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Vertical Asymptote
The line x a is called a vertical asymptote of
the curve y f(x) if at least one of the
following six statements is true
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Examples
Find the limit. 1. 2. State all vertical
asymptotes for the following function and write
the equivalent limit statement for each
asymptote.
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Definition
L
f
Let f be a function defined on some interval (a,
8). Then
means that the value of f (x) can be made as
close to L as we like by taking x sufficiently
large.
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Horizontal Asymptote
The line y L is called a horizontal asymptote
of the curve y f(x) if either
or
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Examples
Evaluate the following. State the equations of
any asymptotes that result from the limit.
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Algebra Review

• Simplify
• Bring the expression into the radical and
  simplify.

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Properties
If n is a positive integer, then
, where a is some constant.
To evaluate limits going to infinity, we often
use the technique of multiplying the expression
by 1 in the form of
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Examples
Find the limit. 1. 2. 3.