# Math Help - Continuity, sequence

1. ## Continuity, sequence

In order to determine the limit of the following sequence perform an algebraic manipulation to express the sequence in a form that is amenable to continuity arguments:

$a_n := \sqrt{n^2+n} -n$

----------

Hmm... Now, I suspect we need to make a substitution, eg.

$n = \frac{1}{x}$

so that as $n \to \infty,~ x \to 0$

So we have $a_x = \sqrt{\frac{1}{x^2}+\frac{1}{x}}-\frac{1}{x} = \dots$ which I got down to:

$\frac{\sqrt{1+x}-1}{x}$

but I need it to be defined at x=0 for continuity to work.

2. $\sqrt {n^2 + n} - n = \frac{n}
{{\sqrt {n^2 + n} + n}} = \frac{1}
{{\sqrt {1 + \frac{1}
{n}} + 1}}$

3. I'm sorry, you've lost me.

How did you get from the first to the second? And then the second to the third?

My algebra must be rusty...

4. $1^{st}\rightarrow 2^{nd}$ is an algebraic identity $A - B = \frac{A^2 - B^2}{A + B}.$

$2^{nd}\rightarrow 3^{rd}$ is dividing through by $n.$