Limits as x tends to ∞. We shall now return to functions of a continuous real variable. We shall confine ourselves entirely to one-valued functions, and we shall denote such a function by ϕ(x). We suppose x to assume successively all values corresponding to points on our fundamental straight line A, starting from some definite point on the line and progressing always to the right. In these circumstances we say that x tends to infinity, or to ∞, and write x → ∞. The only difference between the ‘tending of n to ∞’ discussed in the last chapter, and this ‘tending of x to ∞’, is that x assumes all values as it tends to ∞, i.e. that the point P which corresponds to x coincides in turn with every point of Λ to the right of its initial position, whereas n tended to ∞ by a series of jumps. We can express this distinction by saying that x tends continuously to ∞.

As we explained at the beginning of the last chapter, there is a very close correspondence between functions of x and functions of n. Every function of n may be regarded as a selection from the values of a function of x. In the last chapter we discussed the peculiarities which may characterise the behaviour of a function ϕ(n) as n tends to ∞. Now we are concerned with the same problem for a function ϕ(x); and the definitions and theorems to which we are led are practically repetitions of those of the last chapter.