Limits of Sequences: a,b>0, n→∞

[cauchy21]

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

 

[HallsofIvy]

(a) lim_{n\rightarrow\infty} (\sqrt{(n + a)(n + b)} - n) where a, b > 0

Rationalize the numerator by multiplying both numerator and denominator by \sqrt{n+ a)(n+b)}+ n. Then divide both numerator and denominator by n.

[/quote](b)lim _{n\rightarrow\infty} (n!)^1/n2[/QUOTE]
If y= (n!)^{1/n^2} then
ln(n)= \frac{ln(n!)}{n^2}= \frac{ln(2)}{n^2}+ \frac{ln(3)}{n^2}+ \cdot\cdot\cdot+ \frac{ln(n)}{n^2}.