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

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: ${\lim}_{x \to 2} \left(2 x - 3\right) = 1$
For all $\epsilon > 0$, there exists $\delta = \frac{\epsilon}{2} > 0$ such that
$0 < | x - 2 | < \delta R i g h t a r r o w | x - 2 | < \frac{\epsilon}{2} R i g h t a r r o w 2 | x - 2 | < \epsilon$
$R i g h t a r r o w | 2 x - 4 | < \epsilon R i g h t a r r o w | \left(2 x - 3\right) - 1 | < \epsilon$
Hence, ${\lim}_{x \to 2} \left(2 x - 3\right) = 1$.