Question

Using the $\in -\delta$ definition prove that $\underset{x\to 1}{lim}(2x-1)=1$

Answer 1

The function $f\left(x\right)=2x-1$ is a polynomial and as such it is continuous for every $x\in R$. Then $\underset{x\to 1}{lim}f\left(x\right)=f\left(1\right)=2\times 1-1=2-1=1$

To prove it by using the definition of limit,

$|f\left(x\right)-1|=|2x-1-1|=2|x-1|$

For $x\in (1-f,1+f)$ with $f>0$, we have

$|f\left(x\right)-1|=2|x-1|<2f$

Given any $\in >0$, choose $f_{\in }<\in /2$ s.t

$x\in (1-f_{\in },1+f_{\in })$

$\Rightarrow |f\left(x\right)-1|<2f_{\in }<\in$

which proves the result.