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

[WantToBeSmart]

Homework Statement

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. Homework Equations which I can't get my head around

$$lim_{k\to\infty}((1+\frac{x^2}{k})^k)^{-0.5} = e^{-\frac{x}{2}}$$

The Attempt at a Solution

I have proved $$lim_{k\to\infty}((1+\frac{x^2}{k})^{-0.5}=1$$, 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.

 

[SammyS]

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

BTW: Either you should have x² in your exponent, or you should have x, rather than x² in your limit expression.

 

[WantToBeSmart]

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

BTW: Either you should have x² in your exponent, or you should have x, rather than x² in your limit expression.

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!

$$lim_{k\to\infty}(1+\frac{b}{k})^k= e^b$$

You're right. I added ^2