Using the definition of limit to prove one

Question

I have doubts to prove that $lim_{x\to1} (2x-1) = 1$ with the definition of $\delta$ and $\epsilon$. I have operated but I do not know how to go on.

$|(2x-1)-(1)|=|2x-2|$

We select $\delta=\epsilon$

$0<|2x-2|<\epsilon$

I checked other answers already posted in the web, but I do not understand this last step. How do you know that inequality is true? And how do I continue proving the limit?

Answer 1

You can just factor out the $2$ to get $$|2x-2|=2|x-1|<2\delta.$$ Set $\delta = \frac {\epsilon}{2} $ and you are done.

What you found with $\delta = \epsilon $ can obviously not work because in the definition of this limit it says $|x-1|<\delta $ which is why you would only get $|2x-2|<2\epsilon $ by setting $\delta=\epsilon $.

Answer 2

Let $\epsilon>0$.

We need $$|2x-1-1|<\epsilon$$ or $$|x-1|<\frac{\epsilon}{2}.$$

Thus, for all $\epsilon>0$ we have $$0<|x-1|<\frac{\epsilon}{2}\Rightarrow|2x-1-1|<\epsilon,$$ which says that $$\lim_{x\rightarrow1}(2x-1)=1$$ by the definition of the limit.