Question

How do you find the limit using the epsilon delta definition?

Answer 1

The `epsilon, delta`-definition can be used to formally prove a limit; however, it is not used to find the limit. Let us use the epsilon delta definition to prove the limit: `lim_{x to 2}(2x-3)=1`

Proof

For all `epsilon>0`, there exists `delta=epsilon/2>0` such that
`0<|x-2| < deltaRightarrow |x-2| < epsilon/2 Rightarrow 2|x-2|< epsilon`
`Rightarrow |2x-4| < epsilon Rightarrow|(2x-3)-1| < epsilon`

Hence, `lim_{x to 2}(2x-3)=1`.