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:

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

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

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

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

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

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

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