limit

It is important to realize that this limit may not equal f(a); indeed, f(a) may not necessarily be defined.

More precisely, f(x) tends to l as x tends to a, written f(x) → l, as x → a, if, given any ε > 0 (however small), there exists δ > 0 (which may depend on ε) such that, for all x, except possibly a itself, lying between a−δ and a + δ, f (x) lies between l−ε and l + ε.

Notice that a itself may not be in the domain of f. For example, let f be the function defined by

Then 0 is not in the domain of f, but it can be shown that

If a is in the domain of f, the f is continuous (see continuous function) at a if the limit of f at a is f(a).

In the above, l is a real number. We write f(x) → ∞ as x → a if, given any K (however large), there is a positive number δ (which may depend on K) such that, for all x, except possibly a itself, lying between a−δ and a + δ, f(x) is greater than K. For example, 1/x2 → ∞ as x → 0. There is a similar definition for f(x) →−∞ as x → a.

If f:M → N is a function between metric spaces, we write that f has limit l if given any ε > 0, there exists δ > 0 such that whenever 0

See algebra of limits.

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