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.
