Limits of Sequences: Manipulating Equations for Standard Limits

[tedwillis]

Homework Statement

Have a few limits that I'm stuck on:

a) lim n->infinity (n(n+1)^(n+1))/(n+2)^(n+2))

b) lim n->infinity (n^n/(n+3)^(n+1))

c) lim n->infinity n^(-1)^n

I've tried my best to understand what to do solve these, but can't get it. We've been given answers to standard limits, so I think we need to manipulate the above equations into a form where we can use standard limits.

Homework Equations

Answers are:
a) 1/e

b) 0

c) infinity

The Attempt at a Solution

 

[haruspex]

a) lim n->infinity (n(n+1)^(n+1))/(n+2)^(n+2))

(n+1)^k = n^k(1+1/n)^k. Apply that top and bottom. Do any of your standard limits look like (1+1/n)^k (where k perhaps depends on n)?

 

[christoff]

There are several ways to interpret each of those sequences, largely due to them having so many exponents. You should write them up in LaTeX. Once you do that, I'll give it a go and see if I can give you a push in the right direction.

What "standard limits" do you have?