# Is this weird limit true?

1. Dec 21, 2011

### nhmllr

Here's the weird limit:

lim n$\rightarrow$ $\infty$ aCn/na = 1/(a!)

Don't ask how I thought this up, but let me explain my reasoning.
Let's say that a=3. Then aCn = (n)(n-1)(n-2)
And because n is approaching infinity, could it be (n)(n)(n) = n3?

I wouldn't normally think of things like this but if it's true then it helps me out.

The only thing is that (n)(n-1)(n-2)= n3 +something*n2+something*n

And when I graphed it it looked like it approaches zero.
I've never learned limits formally. Thanks.

EDIT: aCn is a Choose n, or n!/(a![n-a]!)
EDIT: Wolfram alpha is amazing http://www.wolframalpha.com/input/?i=lim+(x!/(3!(x-3)!x^3))+as+x->+infinity
Sorry for not trying this before posting. I'd still be interested in other replies as to why this is true, though

Last edited: Dec 21, 2011
2. Dec 22, 2011

### HallsofIvy

No, because n is not fixed, it is, as you said, approaching infinity.
In any case, aCn= n(n-1)(n-3)/3!, not just n(n-1)(n-2). That goes to infinity as n goes to infinity.

Now, this is completely different- you have an $n^3$ in the denominator.
$$\frac{_nC_3}{n^3}= \frac{n!}{3!(n-3)!}\frac{1}{n^3}= \frac{n(n-1)(n-2)}{6n^3}$$
Since both numerator and denominator are of third power in n, that limit is 1/6.

Last edited by a moderator: Dec 22, 2011
3. Dec 22, 2011

### aesir

I have not tried yet, but a stirling's approximation of ${}_aC_n$ would give the correct limit
$$n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n$$

4. Dec 24, 2011

### identity1

No, you're limit is correct. It's also not that weird. Consider the expansion of ${n\choose a}=\frac{n(n-1)\ldots (n-a+1)}{a!}$. Then, consider that $\frac{n(n-1)\ldots (n-a+1)}{n^a}=1\cdot (1-\frac{1}{n})\cdot\ldots\cdot (1-\frac{a-1}{n})$. Each of those factors goes to 1 as $n\rightarrow\infty$, thus the whole product goes to 1 as well. The expression ${n\choose a}n^{-a}$ is the same thing, but divided by $a!$.