Just read the "epsilon-delta proofs" topic

Just read the "For those that never learnt well the epsilon-delta proofs" topic in this forum and through it I learned a lot of the proofs I never learned in class properly. However, I have a question. In the homework problem sets there are questions like prove the limit does not exist using the delta-epsilon definition. How does one do this?

For example, how would I show that the limit as x approaches 2 of 2x+1 is not 4?

Re: Just read the "epsilon-delta proofs" topic

Quote:

Originally Posted by

**Barthayn** For example, how would I show that the limit as x approaches 2 of 2x+1 is not 4?

Choose $\displaystyle \epsilon=1$ . If $\displaystyle \lim_{x\to 2}(2x+1)=4$ there exists $\displaystyle \delta>0$ such that $\displaystyle |2x+1-4|=|2x-3|<1$ if $\displaystyle |x-2|<\delta$ . But $\displaystyle x=2+\delta/2$ satisfies $\displaystyle |x-2|<\delta$ .

That is, $\displaystyle |2x-3|=|4+\delta -3|=1+\delta<1$ (contradiction) .