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Simple limit question. need a little help please

  1. Apr 4, 2007 #1
    i need someone to coment on this:

    lim(a^x_n)^x'_n, when n->infinity = (a^lim x_n)^lim x'_n , n-> infinity,

    what i am asking here is if we can go from the first to the second??? or if these two expressions are equal??

    any help??
     
  2. jcsd
  3. Apr 4, 2007 #2

    HallsofIvy

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    Yes, because the exponential function, ax, is continuous.
     
  4. Apr 4, 2007 #3
    one more thing here. Is there a theorem or a deffinition that supports similar expressions in a more generalized way?? i forgot to mention this also
     
  5. Apr 4, 2007 #4

    HallsofIvy

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    I'm not sure what you mean. I was referring to the general fact that, from the definition of "continuous", if xn is a sequence of numbers converging to a and f is a function continuous at a, then
    [tex]\lim_{n \rightarrow \infnty} f(x_n)= f(\lim_{n\rightarrow \infty} x_n)= f(a)[/tex]
     
  6. Apr 4, 2007 #5

    yeah this is what i am asking. But what i want to know is if there is a theorem that states this, what you wrote. Or how do we know that this is so?
     
  7. Apr 5, 2007 #6
    Maybe the squeeze theorem?

    glenn
     
  8. Apr 6, 2007 #7
    As Halls indicated, it's essentially the definition of continuity. The definition of continuity for a function f of one real variable defined on an interval (a,b) is for any x in (a,b),

    [tex]\lim_{y \rightarrow x} f(y) = f(x)[/tex]

    (ie. the limit exists and is equal to f(x))
     
  9. Apr 6, 2007 #8

    Yeah, i know the definition of continuity, i was just wondering if there is a specific theorem that states this, as i have not encountered one on my calculus book. However, i do understand it now.
    Many thanks to all of you.
     
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