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Homework Help: Properties of limits

  1. Jun 16, 2010 #1
    1. The problem statement, all variables and given/known data

    What property of limits says that lim 2^(1/n) = 2^lim (1/n) = 2^0 = 1? Thanks.

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 16, 2010 #2

    Dick

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    It's not a property of limits. It's a property of the function f(x)=2^x. f(x) is continuous at x=0.
     
  4. Jun 16, 2010 #3
    I don't follow that. Because 2^x is continuous at x = 0, this means that lim 2^x = 2 ^ lim x?
     
  5. Jun 16, 2010 #4

    Dick

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    f(x) is continuous at x=a means lim x->a f(x)=f(a). That's the definition of continuity. Apply it to f(x)=2^x and a=0.
     
  6. Jun 16, 2010 #5
    Oh, duh... Thank you so much.
     
  7. Jun 16, 2010 #6

    HallsofIvy

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    But there is a "law of limits" involved:

    If [itex]\lim_{x\to a} f(x)= L[/itex] and [itex]\lim_{n\to\infty} x_n= a[/itex] then [itex]\lim_{n\to\infty} f(x_n)= L[/itex].
     
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