Question

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

Answer 1

`\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`.