Question

lim(n->inf) [sqrt(n)*(sqrt(3n+1) - sqrt(3n))]

any help is appreciated!

Thanks.

Answer 1

distribute
\[\sqrt{3n^{2}+n}-\sqrt{3n^{2}}\]
\[= \sqrt{3n^{2}+n} -\sqrt{3} n\]
sub n = 1/u , change limit to u->0
\[\lim_{u \rightarrow 0} \sqrt{\frac{3}{u^{2}}+\frac{1}{u}}- \frac{\sqrt{3}}{u}\]
simplify
\[\lim_{u \rightarrow 0} \frac{\sqrt{3+u} -\sqrt{3}}{u}\]

Answer 2

this results in 0/0
you can now either use L'hopitals rule or multiply top/bottom by conjugate

Answer 3

use the definition of the derivative

Answer 4

that is what you have

Answer 5

\[f(x)=\sqrt{x}\]

the limit is \[f'(3)\]

Answer 6

@dumbcow, @zarkon can i ask how you knew to look for that relationship?
I see it now, obviously, after you pointed it out.. but I would have never seen that otherwise.

Answer 7

also, i wish i could give both of you the medal, but since dumbcow was here first, it's only fair.. sorry zarkon!

Answer 8

I'll live ;)