How to find the bounds of this siquence

[transgalactic]

(n^2+2)^0.5 - (n)^0.5

i thought of doing a limit where n->infinity

but here i get undefined form and even if i whould get some finite limite
it will only be one bound

and i can't do limit n->-infinity because its a sequence must be positive??

 

[Mark44]

It's not bounded, so you're going to have a tough time finding a bound for it. Of the two terms, the first is dominant and is approximately n. n grows large more quickly than sqrt(n).

$$\lim_{n \rightarrow \infty} \sqrt{n^2 + 2} - \sqrt{n} = \infty$$

 

[JG89]

Factor out a $$\sqrt(n^2)$$ to see that it increases beyond all positive bounds.

$$\lim_{n \rightarrow \infty} \sqrt(n^2 +2) - \sqrt(n) = \lim_{n \rightarrow \infty}\sqrt(n^2)(\sqrt(1 + 2/n^2) - \sqrt(\frac{1}{n}))$$

In the second bracket, the first term converges to 1 and the second term converges to 0. So we now have $$\lim_{n \rightarrow \infty} \sqrt(n^2)*1 = \infty$$