Definitions

# One-sided limit

In calculus, a one-sided limit is either of the two limits of a function f(x) of a real variable x as x approaches a specified point either from below or from above. One writes either

$lim_\left\{xto a^+\right\}f\left(x\right) mathrm\left\{or\right\} lim_\left\{xdownarrow a\right\},f\left(x\right)$

for the limit as x approaches a from above (or "from the right"), and similarly

$lim_\left\{xto a^-\right\}f\left(x\right) mathrm\left\{or\right\} lim_\left\{xuparrow a\right\}, f\left(x\right)$

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

$lim_\left\{xto a\right\} f\left(x\right),$

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.

## Examples

One example of a function with different one-sided limits is the following:

$lim_\left\{x rarr 0^+\right\}\left\{1 over 1 + 2^\left\{-1/x\right\}\right\} = 1,$

whereas

$lim_\left\{x rarr 0^-\right\}\left\{1 over 1 + 2^\left\{-1/x\right\}\right\} = 0.$

Another example is the piecewise function

$f\left(x\right)=left\left\{begin\left\{matrix\right\}x^2 & mbox\left\{ for \right\} x< 3 11-\left(x-3\right)^2& mbox\left\{ for \right\} x>3end\left\{matrix\right\}right.$
At the point x0 = 3 the limit from the left is
$lim_\left\{xrarr 3^-\right\} f\left(x\right) = 9$
while the limit from the right is
$lim_\left\{xrarr 3^+\right\} f\left(x\right) = 11.$
Since these two limits are not equal, this function is said to have a jump discontinuity at x0.

## 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.

## Abel's theorem

A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem.