is within a given tolerance of , provided is within a corresponding tolerance of . These two tolerances are often denoted, respectively, by (the tolerance in the value of ) and (the corresponding tolerance in ). The value of the function at is usually omitted from the approximation; for example, in many cases where limits are useful, the function has no value at (it is undefined there).

-definition of limit

Suppose is a function defined on the real line, and there are two real numbers

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