One-sided Limit

analysis

Definition

A one-sided limit is the limit of a function as the input approaches a specific value from either the left or the right.

Left-sided Limit

Definition

Left-sided Limit

Let be a function. has a left-sided limit at if the function values approach as approaches from the left ().

Equivalently, using the sequential definition:

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Right-sided Limit

Definition

Right-sided Limit

Let be a function. has a right-sided limit at if the function values approach as approaches from the right ().

Equivalently, using the sequential definition:

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Implications of Equality

A function has a two-sided limit at if and only if both one-sided limits exist and are equal:

If the one-sided limits exist but are not equal, the function has a jump discontinuity at .