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Limit question: from proof that t-distribution approaches N(0,1)

  1. Nov 7, 2011 #1
    1. The problem statement, all variables and given/known data

    I'm trying to prove that the random variable with t distribution with k degrees of freedom will have N(0,1) as k→∞ the two equations below are derived from the pdf of the t-distribution:

    2. Relevant equations which I can't get my head around

    [tex]lim_{k\to\infty}((1+\frac{x^2}{k})^k)^{-0.5} = e^{-\frac{x}{2}}[/tex]

    3. The attempt at a solution

    I have proved [tex]lim_{k\to\infty}((1+\frac{x^2}{k})^{-0.5}=1 [/tex], which I admit wasn't too hard.

    The statement involving the limit of gamma function was given to us as "knowing that it can be shown" but I would appreciate if someone was willing to explain it/point me to the source where I can find it.
  2. jcsd
  3. Nov 7, 2011 #2


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    Do you know that [itex]\displaystyle \lim_{t \to \infty } \left ( 1 + \frac{1}{t} \right )^{t}=e\ ?[/itex]

    BTW: Either you should have x2 in your exponent, or you should have x, rather than x2 in your limit expression.
  4. Nov 7, 2011 #3
    Hi Sammy, I know the definition of e but the limit is still not trivial to me. Is the equation below some sort of a rule? Thanks!

    [tex]lim_{k\to\infty}(1+\frac{b}{k})^k= e^b[/tex]

    You're right. I added ^2
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