An initial definition provides an intuitive understanding of limits.

Definition. Suppose f(x) is defined when x is near the number a (This means that f is defined around a, except possibly at a itself.). Then we write limxaf(x)=L and say “the limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L by restricting x to be sufficiently close to a but not equal to a.

When we consider both the left and right limits, we define a limit limxaf(x) to exist if and only if both the left and right limits exist and satisfy the condition: limxaf(x)=limxa+f(x)

The the precise definition of limits uses the language of ε and δ.

Definition. limxaf(x)=L exists if for any ε>0, there exists δ such that $ f(x)-L <\varepsilonwhile x-a <\delta( x-a <\delta\Rightarrow f(x)-L <\varepsilon$).