One-sided limits
LIMITS & CONTINUITY
General vs. one-sided limits
When you hear your professor talking about limits, he or she is usually talking about the general limit. Unless a right- or left-hand limit is specifically specified, you’re dealing with a general limit.
The general limit exists at the point ???x=c??? if
1. The left-hand limit exists at ???x=c???,
2. The right-hand limit exists at ???x=c???, and
3. The left- and right-hand limits are equal.
These are the three conditions that must be met in order for the general limit to exist. The general limit will look something like this:
???\lim_{x\to 2}\ f(x)=4???
You would read this general limit formula as “The limit of ???f??? of ???x??? as ???x??? approaches ???2??? equals ???4???.”
Left- and right-hand limits may exist even when the general limit does not. If the graph approaches two separate values at the point ???x=c??? as you approach ???c??? from the left- and right-hand side of the graph, then separate left- and right-hand limits may exist.
Left-hand limits are written as
???\lim_{x\to 2^-}\ f(x)=4???
The negative sign after the ???2??? indicates that we’re talking about the limit as we approach ???2??? from the negative, or left-hand side of the graph.
Right-hand limits are written as
???\lim_{x\to 2^+}\ f(x)=4???
The positive sign after the ???2??? indicates that we’re talking about the limit as we approach ???2??? from the positive, or right-hand side of the graph.
In the graph below, the general limit exists at ???x=-1??? because the left- and right- hand limits both approach ???1???. On the other hand, the general limit does not exist at ???x=1??? because the left-hand and right-hand limits are not equal, due to a break in the graph.
You can see from the graph that the left- and right-hand limits are equal at ???x=-1???, but not at ???x=1???.