Optimized "Homework Solutions for Hard Limits

[freshman2013]

Homework Statement

1. lim as n approaches infinity of ((n+1)^5-(n-1)^5)/n^4
2. lim as n to infinity (n!)^2/(2n)!

Homework Equations

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.

 

[haruspex]

I try to simplify that down to (n+1)-(n-1)

How did you simplify it to that?
Just expand (n+1)⁵ etc. by the binomial theorem.

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.

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?

 

[pbuk]

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.