Formal Epsilon Delta Definition of a Limit

For every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0<|x-c|<\delta$ then $|f(x)-L|<\epsilon$.

In the $\epsilon$-$\delta$ definition of a limit, the value of $\delta$ is chosen independently of $\epsilon$.

The value of $\delta$ is typically dependent on the chosen $\epsilon$, showing a direct relationship between the two.

The formal definition of a limit ensures that the function’s values can be made arbitrarily close to L as x approaches c.