Question

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

Answer 1

$\displaystyle\lim_{x\to\infty}\frac{x^2+1}{x^2}=1$

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

$x>M\implies\left|\dfrac{x^2+1}{x^2}-1\right|$

Working backwards, we have

$\left|\dfrac{x^2+1}{x^2}-1\right|=\left|1+\dfrac1{x^2}-1\right|=\dfrac1{x^2}$
$\implies\sqrt{\dfrac1{x^2}}$

So, whenever $x>M=\dfrac1{\sqrt\varepsilon}$, we can always guarantee that $\left|\dfrac{x^2+1}{x^2}-1\right|$.