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Introductory Limits question

  1. Sep 19, 2011 #1
    1. The problem statement, all variables and given/known data
    Evaluate
    lim ((2+h)^4 - 16)/h
    h->0



    2. Relevant equations
    Difference of squares


    3. The attempt at a solution
    knowing that the top of the fraction is a difference of squares
    i factored it and arrived at lim h->0 ((2+h)^2+4)((2+h)^2-4)/h
    this is where i got stuck
     
  2. jcsd
  3. Sep 19, 2011 #2

    zcd

    User Avatar

    Try multiplying the numerator expression to simplify instead of factoring it.
     
  4. Sep 19, 2011 #3
    Now just expand both [itex](2+h)^{2}[/itex] as well, and then distribute what's left in the numerator. If you completely simplify the whole thing to expand that giant polynomial at the top, you might be able to cancel that h in the denominator and take the limit.
     
  5. Sep 19, 2011 #4
    well, i think i followed your method quark which was to expand the (2+h)^2
    i ended up (after simplification) with lim h->0 (h^2 + 4h + 8)(h^2 + 4h)/h but i do not know how to expand that. can you please show me how to arrive to the answer?

    the answer in the back of the textbook is 32.

    also, is this the easiest way to solve the problem?
     
  6. Sep 19, 2011 #5
    From:
    [tex]\frac{(h^{2}+4h+8)(h^{2}+4h)}{h}[/tex]
    You can distribute the numerator. It works the same way as if you were going to distribute/foil something like (a+b)(c+d)=(ac+da+bc+dc)

    In this case it would be something more like this:
    (a+b+c)(d+e+f) = (ad+ae+af+bd+be+bf+cd+ce+cf)

    I think the easiest way of solving this problem (without using any calculus other than the limit) would be to expand the ()^4 right from the start. It's important to know how to distribute like shown above though.
     
  7. Sep 19, 2011 #6
    alright, i've arrived at my answer :)
    thank you very much
     
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