Simplify Limit Help: (a) (n + a)(n + b) and (b) n!/n^2 | Positive Numbers"

[cauchy21]

Homework Statement

(a) lim_{n\rightarrow\infty} (\sqrt{(n + a)(n + b)} - n) where a, b > 0
(b)lim _{n\rightarrow\infty} (n!)^1/n2

Homework Equations

The Attempt at a Solution

I tried to use Limit theorems but nothing happened.

 

[jbunniii]

For part (a), try multiplying by

\frac{\sqrt{(n+a)(n+b)} + n}{\sqrt{(n+a)(n+b)} + n} = 1

and simplifying until you can evaluate the limit term-by-term.

For part (b), try defining

y_n = \log x_n

and see if you can find

\lim_{n\rightarrow \infty} y_n

and if so, can you relate that to

\lim_{n\rightarrow \infty} x_n?