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Limits are very important in calculus. They are also closely related to the idea of continuity.
Assume we have a function f which might not be properly defined at a point p. We can define q as the limit of f(x) as x approaches p provided that
There are also notions of "limit approaching from above" where we only consider positive values of h, and "limit approaching from below" where we only consider negative values of h. If the ordinary limit is defined, both the limit from above and the limit from below are also defined, and all three agree.
For arbitrary functions and points, there is no guarantee that a limit exists.
If q is the limit of f(x) as x approaches (positive) infinity, the definition is:
Some examples, giving just the results (proof that they satisfy the rules above left to the reader :-):
If the limit of f(x) as x approaches b is a, we denote this:
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lim
x→bf(x) = a or lim x→b f(x) = a