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