How to prove that a limit is incorrect using epsilon delta definition of a limit?

Question

I am trying to fully understand the Epsilon-Delta Definition of a Limit. I have no problem using it to prove a limit that is correct, but I am having trouble using it to disprove an incorrect limit.

For instance, I don't know how to disprove

lim(x -> 2) 2x = 40 It would help my understanding if someone could show me this (dis)proof.

Answer 1

HINT

Recall that by definition $\forall \epsilon$ $\,\exists \delta$ such that $\forall x\,$, $0<|x-2|<\delta$ $\implies|2x-40|<\epsilon$

then

• take $\epsilon =2$

• $|2x-40|<2 \iff 19<x<21$

• then check the definition

Answer 2

Let $\varepsilon=1$. Then, for every $\delta>0$, there is some $x\in\mathbb R$ such that$$0<|x-2|<\delta\text{ and }|2x-40|\geqslant 1=\varepsilon.$$For instance, take $x\in\bigl(2,2+\min\{1,\delta\}\bigr)$. Then $2<x<3$ and therefore $4<2x<6$. So, $|2x-40|\geqslant1$.