Uniqueness theorem for limits

Statement

Two-sided limit

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

If $\lim _{x\to c}f(x)=L$ and $\lim _{x\to c}f(x)=M$, then $L=M$.

Left hand limit

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

If $\lim _{x\to c^{-}}f(x)=L$ and $\lim _{x\to c^{-}}f(x)=M$, then $L=M$.

Right hand limit

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

If $\lim _{x\to c^{+}}f(x)=L$ and $\lim _{x\to c^{+}}f(x)=M$, then $L=M$.