Question

What is the limit as x approaches ∞ in the gamma function? Does it even exist?

Answer 1

`lim_(x->oo) Gamma(x) = oo`

Explanation:

Let's remember what the Gamma function is defined as.

`Gamma(x) = int_0^oo t^(x-1)e^(-t)dt`

If we take the limit as `x->oo`, we have

`lim_(x->oo) Gamma(x) = lim_(x->oo)int_0^oo t^(x-1)e^(-t)dt = int_0^oo t^oo e^(-t)dt`

As you can see, the integral `color(red)("diverges")`, as `t^oo` must approach infinity for `t>1`.

This means that the Gamma function becomes arbitrarily large as `x ->oo`.

Another way to check this is by assuming `x` to be an integer. This is with little loss of generality.

For integer `k`, we know that

`Gamma(k) = (k-1)! = (k-1)(k-2)(k-3)...*3*2*1`.

If we substitute `x` into the equality, we have

`lim_(x->oo) Gamma(x) = (x-1)! = (oo-1)(oo-2)(oo-3)... = oo`

One final way to see that `Gamma(x) ->oo` is by the graph of the function.

This is what it looks like:

graph{(x-1)! [-10, 10, -1, 10]}

As you can see, it grows without bound.