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Using Algebraic Manipulation to evaluate a limit

  1. Dec 17, 2011 #1
    1. The problem statement, all variables and given/known data

    limx→3 (x2+5x+2) ÷ x2-6x+9


    2. The attempt at a solution

    You cannot solve through direct substitution because the denominator comes out to equal 0, and you cannot divide by 0. Thus, I factored the lower terms to reach the following:

    (x-3)2 or (x-3)(x-3).

    However, to my knowledge I cannot do this to the numerator, as ±1 & ±2 do not add up to 5 in any way, shape, or form. As such, I manipulated (x2+5x+2) into the following:

    x(x+5)+2

    But I am not sure where to go from here as the denominator still adds up to 0, and I am not sure of how to simplify any further. As of right now I have:

    x(x+5)+2 ÷ (x-3)(x-3)
     
  2. jcsd
  3. Dec 17, 2011 #2

    rock.freak667

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    Try dividing the numerator and denominator by x^2 after dividing the polynomial.
     
  4. Dec 17, 2011 #3

    SammyS

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    I assume you mean that you're trying to find limx→3 (x2+5x+2) ÷ (x2-6x+9). (The parentheses are important.)

    Usually, one simply observes that as x→3, the numerator → 2 and the denominator → 0 (from the positive side) so that the limit is +∞ .
     
    Last edited: Dec 17, 2011
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