Limit of a function

A limit is the value that a function’s outputs approach as its input approaches a specified point or tends toward infinity. It describes nearby behavior, which need not match the function’s value at the point.

Epsilon–delta definition: A formal definition of a function limit using bounds on input and output distances. It states exactly how closeness of inputs guarantees closeness of outputs.
Function: A rule assigning each input in a domain exactly one output. A limit studies the outputs assigned to inputs near a chosen location.
Derivative: The instantaneous rate of change of a function, defined by a limit of difference quotients. A derivative exists when nearby average rates approach a definite value.
Removable discontinuity: A discontinuity caused by a missing or mismatched function value where the finite limit exists. It separates the nearby limiting value from the function’s assigned value.
One-sided limit: A limit taken as the input approaches a point from only the left or only the right. Separate approach directions can reveal behavior hidden by a two-sided limit.
Continuity: A property of a function whose value at a point agrees with its limit there. Continuity is precisely the case where limiting behavior matches the actual value.
Riemann integral: An integral defined as the limit of sums of function values over increasingly fine partitions. Its construction uses limits of approximating sums.
Jump discontinuity: A discontinuity where finite left-hand and right-hand limits exist but are unequal. The conflicting one-sided limits prevent a two-sided limit.
Infinite limit: A limit in which function values grow without bound as the input approaches a point. It extends limit language to unbounded outputs near a finite input.
Neighborhood (mathematics): A set containing an open interval around a point, or its analogue in a broader space. Limits concern function behavior arbitrarily close to a point, within its neighborhoods.

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