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:

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

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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.

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

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

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