One basic definition of a limit of a sequence of numbers a_{n} is that : whatever positive delta you choose, there is a number N such that for all bigger numbers n the difference between a_{n} and l (ell) is less than d. This may be written mathematically as: d, $ N, n s N, |a_{n }– l| < d. The idea of limit applies just as well when one has a distance between objects defined rather than the difference between numbers. In this case there is a limiting object.

One says that “the limit of the sequence a_{n} is l (ell)” or “”a_{n} converges to l”. One can also say that “this happens as n tends to infinity”. Any sequence that has a limit is called convergent.

We say that a function f is continuous at a if e > 0, $ d > 0, (| x – a| < d |f(x) – f(a)| < e). As with limits this idea can be extended to more general cases where distance between objects in X and a possibly different distance between objects in Y is known. Then we have f is continuous at a if e > 0, $ d > 0, ( distance(x,a) < d distance(f(x),f(a)) < e). We also say f is continuous if f is continuous at every a. It turns out that continuous functions are functions that conserve the structure of convergent sequences. For a continuous function f, if a_{n }converges to x, a_{n} z x, then f(a_{n}) converges to f(x), f(a_{n}) z f(x).

The derivative of a function f at a value a is the rate of change of f(x) as x passes through a. One may think of this as the linear approximation to f at a. This derivative idea of functions may be generalized to linear maps. If u is a function from R^{n} to R^{m} then then u is said to be differentiable at x Î R^{n} if there is a linear map T: R^{n} to R^{m} such that u(x + h) = u(x) + T(h) + e(h), with e(h) small relative to h. The linear map T is the derivative of u at h.

A special case is with m=1. The matrix of T is a row vector of length n which may be denoted as f(x) and is referred to as the gradient of f at x. This vector points in the direction in which f increases most rapidly and its magnitude is the rate of change in that direction.