CalcNotes0203-page14

CalcNotes0203-page14 - (Section 2.3: Limits and Infinity I)...

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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 fraction separately. 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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This note was uploaded on 12/29/2011 for the course MATH 150 taught by Professor Sturst during the Fall '10 term at SUNY Stony Brook.

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