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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