# A Fraction Has To Be Made

Calculus Level 3

We know that the limit of ratios of the two functions $$\sqrt{x^2+x+1}$$ and $$\sqrt{x^2-x-1}$$ as $$x$$ approaches infinity is equal to 1. That is,

$\displaystyle \lim_{x\to\infty} \dfrac{ \sqrt{x^2+x+1} }{\sqrt{x^2-x-1}} = \lim_{x\to\infty} \sqrt{ \dfrac{1 + 1/x + 1/x^2}{1 - 1/x - 1/x^2} } = 1 .$

Is it also true that the limit of the differences of the two same functions as $$x$$ approaches infinity is also equal to 1? That is, is the following limit true?

$\displaystyle \lim_{x\to\infty} \left ({ \sqrt{x^2+x+1} } - {\sqrt{x^2-x-1}} \right) = 1$

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