Mathematical limit

Mathematical limits are used to describe the behaviour of a function as it gets close to a point where the function is not necessarily defined. Also there is the case of a limit "approaching infinity".

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

For every ε > 0 there exists a δ > 0 such that whenever 0 < |h| < δ (where |h| is the absolute value of h), |q - f(p+h)| < ε.

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:

For every ε > 0 there exists a β such that whenever x > β, |q - f(x)| < ε.

Some examples, giving just the results (proof that they satisfy the rules above left to the reader :-):

• The limit as x approaches infinity of 1/x is 0.
• The limit as x approaches 3 of x² is 9. (In this case the value of the function happens to be well defined at the point, and the value is the same as the limit.)
• The limit as x approaches 0 of x^x is 1.
• The limit as x approaches 0 of ((p + x) · (p + x) - p² ) / x is 2 · p.

If the limit of f(x) as x approaches b is a, we denote this:

lim x→b

f(x) = a

or

lim x→b f(x) = a

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