ABSTRACT
In this example, the members of the sequence were functions, so we are not surprised to discover that the limit is also a function f(x) = x. We would like to adapt the definition of the limit to this new situation. Let us look carefully at Definition 1.2.5. A number L qualified as a limit of the sequence {an} if the members of {an} were as close to L as needed, and this distance was measured on the number line. For example, the fact that lim 1/n = 0 is intuitively clear: if we replace n by a large number, say n = 1000, then 1/1000 = 0.001 is close to 0. If we apply the same reasoning to the sequence {fn}, we should have f1000 “close to” f . However, f1000(x) = (1000x)/(1001 + x
2), so we need to clarify in what sense this function is close to f(x) = x.
