, a one-sided limit
is either of the two limits
of a function f
) of a real
approaches a specified point either from below or from above. One writes either
for the limit as x approaches a from above (or "from the right"), and similarly
for the limit as x approaches a from below (or "from the left").
The two one-sided limits exist and are equal if and only if the limit of f(x) as x approaches a exists. In some cases in which the limit
does not exist, the two one-sided limits nonetheless exist. Consequently the limit as x approaches a is sometimes called a "two-sided limit". In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.
One example of a function with different one-sided limits is the following:
Another example is the piecewise function
At the point x0
= 3 the limit from the left is
while the limit from the right is
Since these two limits are not equal, this function is said to have a jump discontinuity
Relation to topological definition of limit
The one-sided limit to a point p
corresponds to the general definition of limit
, with the domain of the function restricted to one side, by either allowing that the function domain is a subset of the topological space, or by considering a one-sided subspace, including p
A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.