One-sided limit

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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

$\lim_{x\to a^+}f(x)\ \mathrm{or}\ \lim_{x\downarrow a}\,f(x)$

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

$\lim_{x\to a^-}f(x)\ \mathrm{or}\ \lim_{x\uparrow a}\, f(x)$

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

$\lim_{x\to a} f(x)\,$

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.

1 Examples
2 Relation to topological definition of limit
3 Abel's theorem
4 See also
5 External links

[edit] Examples

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

The piecewise function $f(x)=\left\{\begin{matrix}x^2 & \mbox{ for } x< 3 \\ 11-(x-3)^2& \mbox{ for } x>3\end{matrix}\right.$

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

whereas

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

Another example is the piecewise function

$f(x)=\left\{\begin{matrix}x^2 & \mbox{ for } x< 3 \\ 11-(x-3)^2& \mbox{ for } x>3\end{matrix}\right.$

At the point x₀ = 3 the limit from the left is

$\lim_{x\rarr 3^-} f(x) = 9$

while the limit from the right is

$\lim_{x\rarr 3^+} f(x) = 11.$

Since these two limits are not equal, this function is said to have a jump discontinuity at x₀.

[edit] 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.