One-sided limit

In mathematics, a one-sided limit is where the limit of a function is defined in moving in the positive or negative direction, but not both.

For example, consider the function

$f(x) = \left\{ \begin{matrix} 1\ \mbox{if}\ x\isin (-\infty, 1] \\ 0\ \mbox{if}\ x\isin (1, \infty) \\ \end{matrix}\right.$

Now, if we take the limit moving from smaller values of x to 1, noted,

$\lim_{x\rarr 1} f(x) = 1$

but if we take the limit moving from larger values of x to 1, noted,

$\lim_{x\rarr 1^{-}} f(x) = 0$

These two limits are not identical and so f has a one-sided limit at the point x=1.

If these two limits are identical, f is said to have a two-sided limit at that point.

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07-14-2008 23:18:10
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