One basic definition of a limit of a sequence of numbers an is that : whatever positive delta you choose, there is a number N such that for all bigger numbers n the difference between an and l (ell) is less than d. This may be written mathematically as: 
d, $ N, n s N, |an – 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 an is l (ell)” or “”an 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 an converges to x, an z x, then f(an) converges to f(x), f(an) 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 Rn to Rm then then u is said to be differentiable at x Î Rn if there is a linear map T: Rn to Rm 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.