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Homework Help: Essentials of Calculus: find the limit problem

  1. Jul 25, 2010 #1
    1. The problem statement, all variables and given/known data

    SYNTHESIS

    Find limit as x approaches 3 of : (x^3-27)/x-3

    2. Relevant equations



    3. The attempt at a solution

    I typed it into my calculator and got x^3 - (27)/(x-3) where x^3 wasn't divided by x-3. I really do not know where to start. I know the answer is 27 from the back of the book.

    I was thinking it was something like x^3- (3)^3/ (x-3) and then you can cancel out one of the x-3 but I still cannot get 27 and if you plugin 3 into the original equation you get 0.
     
    Last edited: Jul 25, 2010
  2. jcsd
  3. Jul 26, 2010 #2
    Try factoring x3-27
     
  4. Jul 26, 2010 #3

    HallsofIvy

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    Science Advisor

    You need to know that [itex](a- b)(a^2- ab+ b^2)= a^3- b^3[/itex].
     
  5. Jul 26, 2010 #4
    Or, if you ever forget that a³ - b³ = (whatever it actually equals :tongue2: )
    It looks like something that can be factored.

    Set x³ - 27 equal to zero.

    x³ - 27 = 0

    Find an x value that makes x³ - 27 = 0 hold, i.e. x = 3

    (3)³ - 27 = 0

    Then, because x = 3 is an answer, x - 3 = 0 is a factor so divide x - 3 into x³ - 27

    ______x²_+_..._______​
    x - 3 |x³ + 0x² + 0x - 27
    x³ - 3x²​
    ....​

    Keep going & you'll factor it & get your answer :wink:
     
  6. Jul 26, 2010 #5

    hunt_mat

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    Homework Helper

    Or failing that L'Hopital's rule...

    Mat
     
  7. Jul 26, 2010 #6
    Or use a substitution with u = x - 3, no factoring cubes or l'Hôpital's rule. :wink:
     
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