CalcNotes0203-page25

CalcNotes0203-page25 - stays nonnegative” as x ± ² , in...

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(Section 2.3: Limits and Infinity I) 2.3.25 When taking the limit of an algebraic function, Dominant Term Substitution may also be applied to radicands and bases of powers, provided there is no problem with expressions being defined, and there are no “ties” as in Example 20. A radical is a kind of power. For example, x = x 1/ 2 . (See Footnotes 8 and 9 on how dominance can fail us for non-algebraic functions.) Example 19 Evaluate lim x ±² 4 x 3 ³ x 10 ³ 5 x + 3 () 2 . Solution Method (The “Short Cut”: Dominant Term Substitution) Note : Although the radicand, x 10 ± 5 , can be negative in value, we have a different situation from the previous example. It “eventually
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Unformatted text preview: stays nonnegative” as x ± ² , in the sense that x 10 ± 5 ² on the x-interval c , ± ( ) for some real constant c . Therefore, the radical “eventually” yields real values as x ± ² . In the radicand, x 10 ± 5 , x 10 dominates ± 5 . In the power-base, x + 3 , x dominates 3. “Short Cut” Solution lim x ±² 4 x 3 ³ x 10 ³ 5 x + 3 ( ) 2 = lim x ±² 4 x 3 ³ x 10 x ( ) 2 = lim x ±² 4 x 3 ³ x 5 x 2 = lim x ±² ³ x 5 x 2 = lim x ±² ³ x 3 = ³ ² The Factoring Principle of Dominance should not be applied locally to the radicand, x 10 ± 5 . Example 20 will show how that approach can fail....
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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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