How do I do this Sequences limit question

How do I evaltuate this limit as n goes to infinity?

(n^n)/((n+3)^(n+1))

Please show your working

Thanks, James

What if we factor the denominator (n + 3)^(n + 1) as (n + 3)(n + 3)^n ?

Then, our rational expression becomes:

n^n / ((n + 3)(n + 3)^n)

We can write this as the product of two rational expressions:

(n^n / (n + 3)^n) (1 / (n + 3))

Then, we take the product of the two limits:

lim (n^n / (n + 3)^n) lim (1 / (n + 3))

Now, we don't care what the limit of the first term is, as long as it converges (Note: it's 1/e^3), because we know the limit of the second term lim (1 / (n + 3)) = 0 as n -> infinity.

So, the product of the two limits is (something times) 0, and so the limit itself is 0.