89-92. Limits as $x to infty$ or $x to −infty$

89. Limits as $x$ tends to $\infty$ .

We shall now return to functions of a continuous real variable. We shall confine ourselves entirely to one-valued functions,¹ 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 $Λ$ , 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 $\infty$ , and write $x \to \infty$ . The only difference between the ‘tending of $n$ to $\infty$ ’ discussed in the last chapter, and this ‘tending of $x$ to $\infty$ ’, is that $x$ assumes all values as it tends to $\infty$ , 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 $\infty$ by a series of jumps. We can express this distinction by saying that $x$ tends continuously to $\infty$ .

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 $\infty$ . 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. Thus corresponding to Def. 1 of § 58 we have:

Definition 1. The function $ϕ (x)$ is said to tend to the limit $l$ as $x$ tends to $\infty$ if, when any positive number $ϵ$ , however small, is assigned, a number $x_{0} (ϵ)$ can be chosen such that, for all values of $x$ equal to or greater than $x_{0} (ϵ)$ , $ϕ (x)$ differs from $l$ by less than $ϵ$ , if $| ϕ (x) - l | < ϵ$ when $x \geq x_{0} (ϵ)$ .

When this is the case we may write $lim_{x \to \infty} ϕ (x) = l,$ or, when there is no risk of ambiguity, simply $lim ϕ (x) = l$ , or $ϕ (x) \to l$ . Similarly we have:

Definition 2. The function $ϕ (x)$ is said to tend to $\infty$ with $x$ if, when any number $Δ$ , however large, is assigned, we can choose a number $x_{0} (Δ)$ such that $ϕ (x) > Δ$ when $x \geq x_{0} (Δ)$ .

We then write $ϕ (x) \to \infty .$ Similarly we define $ϕ (x) \to - \infty$ .² Finally we have:

Definition 3. If the conditions of neither of the two preceding definitions are satisfied, then $ϕ (x)$ is said to oscillate as $x$ tends to $\infty$ . If $| ϕ (x) |$ is less than some constant $K$ when $x \geq x_{0}$ ,³ then $ϕ (x)$ is said to oscillate finitely, and otherwise infinitely.

The reader will remember that in the last chapter we considered very carefully various less formal ways of expressing the facts represented by the formulae $ϕ (n) \to l$ , $ϕ (n) \to \infty$ . Similar modes of expression may of course be used in the present case. Thus we may say that $ϕ (x)$ is small or nearly equal to $l$ or large when $x$ is large, using the words ‘small’, ‘nearly’, ‘large’ in a sense similar to that in which they were used in Ch. IV.

Example XXXIV

1. Consider the behaviour of the following functions as $x \to \infty$ : $1 / x$ , $1 + (1 / x)$ , $x^{2}$ , $x^{k}$ , $[x]$ , $x - [x]$ , $[x] + \sqrt{x - [x]}$ .

The first four functions correspond exactly to functions of $n$ fully discussed in Ch. IV. The graphs of the last three were constructed in Ch. II (Exs. XVI. 1, 2, 4), and the reader will see at once that $[x] \to \infty$ , $x - [x]$ oscillates finitely, and $[x] + \sqrt{x - [x]} \to \infty$ .

One simple remark may be inserted here. The function $ϕ (x) = x - [x]$ oscillates between $0$ and $1$ , as is obvious from the form of its graph. It is equal to zero whenever $x$ is an integer, so that the function $ϕ (n)$ derived from it is always zero and so tends to the limit zero. The same is true if $ϕ (x) = \sin x π, ϕ (n) = \sin n π = 0.$ It is evident that $ϕ (x) \to l$ or $ϕ (x) \to \infty$ or $ϕ (x) \to - \infty$ involves the corresponding property for $ϕ (n)$ , but that the converse is by no means always true.

2. Consider in the same way the functions: $(\sin x π) / x, x \sin x π, (x \sin x π)^{2}, \tan x π, a \cos^{2} x π + b \sin^{2} x π,$ illustrating your remarks by means of the graphs of the functions.

3. Give a geometrical explanation of Def. 1, analogous to the geometrical explanation of Ch. IV, § 59.

4. If $ϕ (x) \to l$ , and $l$ is not zero, then $ϕ (x) \cos x π$ and $ϕ (x) \sin x π$ oscillate finitely. If $ϕ (x) \to \infty$ or $ϕ (x) \to - \infty$ , then they oscillate infinitely. The graph of either function is a wavy curve oscillating between the curves $y = ϕ (x)$ and $y = - ϕ (x)$ .

5. Discuss the behaviour, as $x \to \infty$ , of the function $y = f (x) \cos^{2} x π + F (x) \sin^{2} x π,$ where $f (x)$ and $F (x)$ are some pair of simple functions (e.g. $x$ and $x^{2}$ ). [The graph of $y$ is a curve oscillating between the curves $y = f (x)$ , $y = F (x)$ .]

90. Limits as $x$ tends to $- \infty$ .

The reader will have no difficulty in framing for himself definitions of the meaning of the assertions ‘ $x$ tends to $- \infty$ ’, or ‘ $x \to - \infty$ ’ and $lim_{x \to - \infty} ϕ (x) = l, ϕ (x) \to \infty, ϕ (x) \to - \infty .$ In fact, if $x = - y$ and $ϕ (x) = ϕ (- y) = ψ (y)$ , then $y$ tends to $\infty$ as $x$ tends to $- \infty$ , and the question of the behaviour of $ϕ (x)$ as $x$ tends to $- \infty$ is the same as that of the behaviour of $ψ (y)$ as $y$ tends to $\infty$ .

91. Theorems corresponding to those of Ch. IV, §§ 63-67.

The theorems concerning the sums, products, and quotients of functions proved in Ch. IV are all true (with obvious verbal alterations which the reader will have no difficulty in supplying) for functions of the continuous variable $x$ . Not only the enunciations but the proofs remain substantially the same.

92. Steadily increasing or decreasing functions.

The definition which corresponds to that of § 69 is as follows: the function $ϕ (x)$ will be said to increase steadily with $x$ if $ϕ (x_{2}) \geq ϕ (x_{1})$ whenever $x_{2} > x_{1}$ . In many cases, of course, this condition is only satisfied from a definite value of $x$ onwards, i.e. when $x_{2} > x_{1} \geq x_{0}$ . The theorem which follows in that section requires no alteration but that of $n$ into $x$ : and the proof is the same, except for obvious verbal changes.

If $ϕ (x_{2}) > ϕ (x_{1})$ , the possibility of equality being excluded, whenever $x_{2} > x_{1}$ , then $ϕ (x)$ will be said to be steadily increasing in the stricter sense. We shall find that the distinction is often important (cf. §§ 108-109).

The reader should consider whether or no the following functions increase steadily with $x$ (or at any rate increase steadily from a certain value of $x$ onwards): $x^{2} - x$ , $x + \sin x$ , $x + 2 \sin x$ , $x^{2} + 2 \sin x$ , $[x]$ , $[x] + \sin x$ , $[x] + \sqrt{x - [x]}$ . All these functions tend to $\infty$ as $x \to \infty$ .

Thus $\sqrt{x}$ stands in this chapter for the one-valued function $+ \sqrt{x}$ and not (as in § 26) for the two-valued function whose values are $+ \sqrt{x}$ and $- \sqrt{x}$ .↩︎
We shall sometimes find it convenient to write $+ \infty$ , $x \to + \infty$ , $ϕ (x) \to + \infty$ instead of $\infty$ , $x \to \infty$ , $ϕ (x) \to \infty$ .↩︎
In the corresponding definition of SecNo 62, we postulated that $| ϕ (n) | < K$ for all values of $n$ , and not merely when $n \geq n_{0}$ . But then the two hypotheses would have been equivalent; for if $| ϕ (n) | < K$ when $n \geq n_{0}$ , then $| ϕ (n) | < K^{'}$ for all values of $n$ , where $K^{'}$ is the greatest of $| ϕ (1) |$ , $| ϕ (2) |$ , …, $| ϕ (n_{0} - 1) |$ and $K$ . Here the matter is not quite so simple, as there are infinitely many values of $x$ less than $x_{0}$ .↩︎

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Chapter IV

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93-97. Limits as

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89. Limits as x tends to ∞.

90. Limits as x tends to −∞.

91. Theorems corresponding to those of Ch. IV, §§ 63-67.

92. Steadily increasing or decreasing functions.

89. Limits as $x$ tends to $\infty$ .

90. Limits as $x$ tends to $- \infty$ .