Question

What is the limit of `(x-1)(x-2)(x-3)(x-4)(x-5)/(5x-1)^5` as x goes to infinity?

Answer 1

You can separate this into multiple limits using their multiplicative properties.

`= lim_(x->oo) (x-1)/(5x - 1) * lim_(x->oo) (x-2)/(5x - 1) * lim_(x->oo) (x-3)/(5x - 1) * lim_(x->oo) (x-4)/(5x - 1) * lim_(x->oo) (x-5)/(5x - 1)`

Then, we can "simplify" this down using the fact that as `x` becomes really large, the evaluation `(x)/(5x) ~~ 1/5` on each limit becomes more and more accurate (with `x` >> `1`, the `1` becomes insignificant at large `x`).

Since there are five of these limits, you get

`(1/5) * (1/5) * (1/5) * (1/5) * (1/5)`

`= 1/(5^5)`, or `color(blue)(1/(3125))`.