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What is

$$\lim_{x\to 1}\frac{\sqrt{x^2-1}-\sqrt{x-1}}{\sqrt{x^3-1}-\sqrt{x-1}}\,?$$ I'm in secondary school and they throw this. We haven't seen l'hôpital yet but it doesn't work anyway. The answer is $\frac{1}{2} \left(\sqrt{2}-1\right) \left(1+\sqrt{3}\right)$ according to Wolfram but it doesn't show how to solve it. (Step-by-step solution unavailable). Haven't seen complex numbers either but I grasp the basics. Teacher is no help and I used lots of paper and time to be desperate enough to ask here.

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Try multiplying top and bottom by $\sqrt{x^3-1}+\sqrt{x-1}$. – icurays1 Jan 30 '13 at 16:04
up vote 5 down vote accepted

Write $x^2-1=(x-1)(x+1)$ and $x^3-1=(x-1)(x^2+x+1)$. Then divide the numerator and denominator of the fraction by $\sqrt {x-1}$.

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damn now it looks so easy. Thanks! – user60258 Jan 30 '13 at 16:06

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