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.