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G prove that limx→∞(x 2 + 1)/x2 = 1 using the definition of limit at ∞.

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\displaystyle\lim_(x\to\infty)(x^2+1)/(x^2)=1

means to say there is some number
M for which any choice of
\varepsilon>0 such that


x>M\implies\left|(x^2+1)/(x^2)-1\right|<\varepsilon

Working backwards, we have


\left|(x^2+1)/(x^2)-1\right|=\left|1+\frac1{x^2}-1\right|=\frac1{x^2}<\varepsilon

\implies\sqrt{\frac1{x^2}}<\sqrt\varepsilon\implies\frac1{\sqrt\varepsilon}<|x|

So, whenever
x>M=\frac1{\sqrt\varepsilon}, we can always guarantee that
\left|(x^2+1)/(x^2)-1\right|<\varepsilon.
User Matthias Huschle
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