Definitions

one side

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_{xto a^+}f(x) mathrm{or} lim_{xdownarrow a},f(x)

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

lim_{xto a^-}f(x) mathrm{or} lim_{xuparrow 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_{xto 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.

Examples

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

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>3end{matrix}right.
At the point x0 = 3 the limit from the left is
lim_{xrarr 3^-} f(x) = 9
while the limit from the right is
lim_{xrarr 3^+} f(x) = 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.

See also

External links

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