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Unformatted text preview: (Section 2.3: Limits and Infinity I) 2.3.14 How can we tell if a limit is or ? A key tool is something we will call Dominant Term Substitution.
When analyzing a “long-run” limit, if a and b are integers, then
x a dominates x b
a > b.
(See Footnotes 6 and 7 for more on dominance.)
When analyzing a “long-run” limit,
the dominant term of a polynomial is the leading term.
Dominant Term Substitution
The limit of an expression is the same as the limit of its
dominant term. (By “limit” here, we include and
Note: Do not use dominance if an expression is not defined
as a real quantity when considering the limit. This is never
an issue with polynomials.
Why does this work? The Factoring Principle of Dominance:
If the dominant term is factored out of an expression,
we obtain the dominant term times something approaching 1.
The “long-run” limit of the expression is therefore the limit of
the dominant term, and the factor approaching 1 can be
removed when figuring out the “final” limit. This procedure
can be applied to the numerator and the denominator of a
We will see this in Example 12.
Example 12 (A Question of Dominance) ( Evaluate lim x 8
x ) x6 . Solution Method
There is a tension between the two terms, x 8 and x 6 , because
x 8 approaches as x
, while x 6 approaches
(Review graphs of polynomials such as these in Precalculus.)
Is someone in charge here? In the long run, yes! The dominant
term here is x 8 . ...
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