Evaluating a couple of limits

I wonder if someone could clarify these limits for me. They arise in the derivation of the poisson distribution from the binomial distribution as and

The first is

as .

I don't see how this works. If anything I would have thought that

as .

The second is that

\frac{n!}{(n-k)!(n-\lambda)^k = \frac{n(n-1) \ldots (n-k+1)}{(n - \lambda) (n-\lambda) \ldots (n - \lambda)}

is a "quantity that tends to 1 as (since remains constant)."

What precisely is the role of ? Is the critical condition not that the numerator and denominator grow at the same rate, and if so, how can this be demonstrated?

Thanks in advance. MD

Re: Evaluating a couple of limits

Quote:

Originally Posted by

**Mathsdog** If anything I would have thought that

as

.

. Rather, using the Maclaurin series for ln, as .

Quote:

Originally Posted by

**Mathsdog**
is a "quantity that tends to 1 as

(since

remains constant)."

If k is constant, then . Each of these limits is 1. Indeed, as .

Re: Evaluating a couple of limits

Quote:

Originally Posted by

**Mathsdog** I wonder if someone could clarify these limits for me. They arise in the derivation of the poisson distribution from the binomial distribution as

and

The first is

as

.

I don't see how this works. If anything I would have thought that

as

.

The second is that

\frac{n!}{(n-k)!(n-\lambda)^k = \frac{n(n-1) \ldots (n-k+1)}{(n - \lambda) (n-\lambda) \ldots (n - \lambda)}

is a "quantity that tends to 1 as

(since

remains constant)."

What precisely is the role of

? Is the critical condition not that the numerator and denominator grow at the same rate, and if so, how can this be demonstrated?

Thanks in advance. MD