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I need to find the limit of the following sequence:

$$\lim_{x\to \infty} \sqrt{x^2+x}-\sqrt{x^2-x}$$

I can transform the above to:

$$\frac{2x}{\sqrt{x^2+x}+\sqrt{x^2-x}}$$

But I can't seem to prove that the term is diminishing and that its superlum is $1$ (which would prove the limit). Am I going about this completely the wrong way?

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    You're going about it the right way. You can, for instance, divide top and bottom by $x$, while factoring in $x$ inside the square root at the bottom. – 2017-01-31
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    http://math.stackexchange.com/questions/597177/find-the-limit-displaystyle-lim-x-to-infty-sqrtx2x-sqrtx2-x?rq=1 – 2017-01-31

2 Answers 2

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write your limit in the form $$\frac{2x}{x\left(\sqrt{1+1/x}+\sqrt{1-1/x}\right)}$$

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    Sonhard Graubner Thank you! Duh.... – 2017-01-31
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Hint -

You are on right way.

Take x common from terms in denominator.