# Uniqueness theorem for limits

## Statement

### Two-sided limit

Suppose is a function and is a point such that is defined on both the immediate left and the immediate right of . The **uniqueness theorem for limits** states that *if* the limit of exists at (in the sense of existence as a finite real number) then it is unique. In other words:

If and , then .

### Left hand limit

Suppose is a function and is a point such that is defined on the immediate left of . The **uniqueness theorem for left hand limits** states that *if* the left hand limit of exists at (in the sense of existence as a finite real number) then it is unique. In other words:

If and , then .

### Right hand limit

Suppose is a function and is a point such that is defined on the immediate right of . The **uniqueness theorem for right hand limits** states that *if* the right hand limit of exists at (in the sense of existence as a finite real number) then it is unique. In other words:

If and , then .