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Limit problem

  1. Apr 26, 2016 #1
    1. The problem statement, all variables and given/known data
    If ##a, b \in \{1,2,3,4,5,6\}##, then number of ordered pairs of ##(a,b)## such that ##\lim_{x\to0}{\left(\dfrac{a^x + b^x}{2}\right)}^{\frac{2}{x}} = 6## is

    2. Relevant equations


    3. The attempt at a solution
    So, this is a typical exponential limit.

    ##\lim_{x\to0}e^{\frac{2}{x}.\ln\left(\frac{a^x + b^x}{2}\right)} = 6##

    Using L'Hospital

    ##\lim_{x\to0}e^{2.\frac{2}{a^x + b^x}.(a^x\ln a + b^x\ln b)} = 6##

    This on substituting the limit simplifies to

    ##e^{2.(\ln a + \ln b)} = 6##

    ##e^{\ln {(ab)}^2} = 6 \Rightarrow {(ab)}^2 = 6 ## However, the answer only has ##ab = 6##. What's wrong?
     
  2. jcsd
  3. Apr 26, 2016 #2

    PeroK

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    You can't use L'Hopital's rule there!

    That said, I see you're using it inside a continuous function, which I guess is a generalisation of L'Hopital. You just made a simple error with your differentiation.
     
    Last edited: Apr 26, 2016
  4. Apr 26, 2016 #3
    Oh, yesss, I see it now. That was rather dumb. Thank you!
     
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