Function Limits IV: The Gory Details Of One-Sided Limits

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Lesson Summary: 
We provide a mathematically rigorous definition of one-sided limits and prove a very important relationship between two-sided and one-sided limits.
Lesson Inputs: 
Function Limits III: The Gory Details
Lesson: 

In the lesson Function Limits II: When Function Limits Don't Exist we discussed how a function limit from the left and from the right must be equal. We introduced the notation of a function limit from left as
\begin{equation}
\lim_{x\to a\,-}\,f(x)=L_l
\end{equation}
and a function limit from the right as
\begin{equation}
\lim_{x\to a\,+}\,f(x)=L_r.
\end{equation}
We referred to these as one-sided limits. We will oftentimes refer to
\begin{equation}
\lim_{x\to a\,}\,f(x)=L
\end{equation}
as a two-sided limit.

Our goal here is to provide the "epsilon-delta" definition of these limits similar to what we did for the function limit and to show that
\begin{equation}
\lim_{x\to a\,}\,f(x)=L
\end{equation}
if and only if
\begin{equation}
\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.
\end{equation}

Our left-hand limit is defined as

\begin{equation}
\lim_{x\to a\,-}\,f(x)=L_l\quad\mbox{means}\quad\left\{\begin{array}{l}\mbox{for every }\,\epsilon\gt 0\,\mbox{there exists }\,\delta\gt 0\,\mbox{ such that }\\\mbox{if }\,a-\delta\lt x\lt a,\,\mbox{then }\,|f(x)-L_l|\lt\epsilon\end{array}\right.
\end{equation}

Left-Sided Limit

Our right-hand limit is defined as

\begin{equation}
\lim_{x\to a\,+}\,f(x)=L_r\quad\mbox{means}\quad\left\{\begin{array}{l}\mbox{for every }\,\epsilon\gt 0\,\mbox{there exists }\,\delta\gt 0\,\mbox{ such that }\\\mbox{if }\,a\lt x\lt a+\delta,\,\mbox{then }\,|f(x)-L_r|\lt\epsilon\end{array}\right.
\end{equation}

Right-Sided Limit

More importantly,
\begin{equation}
\lim_{x\to a\,}\,f(x)=L
\end{equation}
if and only if
\begin{equation}
\lim_{x\to a-}\,f(x)=L\quad\mbox{and}\quad\lim_{x\to a+}\,f(x)=L.
\end{equation}
The use of "if and only if" is very important. It means two things at the same time. First, it means that if the function limit exists then we know that the left and right-sided limits exist, are equal to each other, and have the value of $L$. Second, it means that if the two sided limits exist and are equal to each other then the two-sided function limit exists and is equal to $L$. You may find the proof here.

Finally, we can prove our assertion that if the left and right-sided limits do not equal one another then the function limit does not exist.

See Function Limits VII: Putting It All Together With Useful Examples.