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

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 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.

[edit] Examples

• We have

whereas

• Consider the function

Then, 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, one has a jump discontinuity at x₀.

[edit] 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.

[edit] External links

• One-sided limit on PlanetMath