A function may approach two different limits. One where the variable approaches its limit through values larger than the limit and the other where the variable approaches its limit through values smaller than the limit. In such a case, the limit is not defined but the right and left-hand limits exist.
- When the limx→af(x)=A+limx→af(x)=A+ given the values of f near x to the right of a. This value is said to be the right hand limit of f(x) at a.
- When the limx→af(x)=A−limx→af(x)=A− given the values of f near x to the left of a. This value is called the left hand limit of f(x) at a.
- The limit of a function exists if and only if the left-hand limit is equal to the right-hand limit. limx→a−1f(x)=limx→a+f(x)=Llimx→a−1f(x)=limx→a+f(x)=L
Note: The limit of the function exists between any two consecutive integers.