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Math Help - Continuity, sequence

  1. #1
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    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.
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  2. #2
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    \sqrt {n^2  + n}  - n = \frac{n}<br />
{{\sqrt {n^2  + n}  + n}} = \frac{1}<br />
{{\sqrt {1 + \frac{1}<br />
{n}}  + 1}}
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  3. #3
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    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...
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  4. #4
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    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.
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