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Limit problem

  1. May 18, 2016 #1
    1. The problem statement, all variables and given/known data

    If a and b satisfy ##\lim_{x->0}\frac{\sqrt{ax+b}-5}{x} = \frac{1}{2}##, then a+b equals...

    A. -15
    B. -5
    C. 5
    D. 15
    E. 30

    2. Relevant equations
    L'hospital


    3. The attempt at a solution

    By using L'hospital, I get b=a^2

    Then, I got stuck.. Substituting b=a^2 into the limit equation, but I still can't cancel out the x which is the cause of zero denominator..
    Please help
     
  2. jcsd
  3. May 18, 2016 #2

    blue_leaf77

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    Since ##x \to 0##, you can write ##\sqrt{ax+b} = \sqrt{b} \sqrt{1+\frac{ax}{b}}## and then expand ##\sqrt{1+\frac{ax}{b}}## into power series. You will be able to determine ##b## first thanks to the presence of ##-5## in the numerator.
     
  4. May 18, 2016 #3

    SammyS

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    What must be true for ##\displaystyle \ \frac{\sqrt{ax+b}-5}{x}\ ## if L'Hôpital's rule can be applied? In particular what must be true of ##\displaystyle \ \lim_{x->0}(\sqrt{ax+b}-5) \ ?##
     
  5. May 18, 2016 #4
    It must be zero!
    So, b equals 25

    and a equals 5

    a+b = 30..

    Thanks a lot!
     
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