Limit of √((n+a)(n+b)) - n and (n!)^(1/n²) as n→∞

[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

$$\lim_{n\rightarrow \infty} x_n$$?