The formal definition of a limit is the cornerstone of calculus.
It describes the behavior of a function f(x) as x gets arbitrarily close to a value a, even if the function isn't defined at that exact point.
As shown, lim_{xβa} f(x) = L means that by choosing x sufficiently close to a, we can force f(x) to be as close to L as we want.
Credit to Cauchy for this rigor!