# Really hard limits

1. Dec 10, 2013

### freshman2013

1. The problem statement, all variables and given/known data
1. lim as n approaches infinity of ((n+1)^5-(n-1)^5)/n^4
2. lim as n to infinity (n!)^2/(2n)!

2. Relevant equations

3. The attempt at a solution
1.I split it up, got (((n+1)/n)^4)*(n+1)-(((n-1)/n)^4)*(n-1). I try to simplify that down to (n+1)-(n-1) and got 2 as my answer, since the other portion of the limit evaluates to 1. Wolframalpha, however, gave me 10 as the answer.

2. I never seen a problem involving limits with just factorials, so I just guessed that n factorial squared grows faaster than (2n)! so the answer is infinity but wolframalpha says the answer is 0.

2. Dec 10, 2013

### haruspex

How did you simplify it to that?
Just expand (n+1)5 etc. by the binomial theorem.
In the expansions of each n! on the top and (2n)! on the bottom, do you see which terms will cancel? What will that leave?

3. Dec 10, 2013

### MrAnchovy

1. Expand $(n + 1)^5$ and $(n - 1)^5$.

2. You guessed wrong. But you don't need to guess - look at the ratio of the (n + 1)th term to the nth term, this is always a good place to start.