93-97. Limits as $x\to a$

93. Limits as $x$ tends to $0$.

Let $\varphi (x)$ be such a function of $x$ that $\underset{x\to \mathrm{\infty }}{lim}\varphi (x)=l$, and let $y=1/x$. Then $$\varphi (x)=\varphi (1/y)=\psi (y),$$ say. As $x$ tends to $\mathrm{\infty }$, $y$ tends to the limit $0$, and $\psi (y)$ tends to the limit $l$.

Let us now dismiss $x$ and consider $\psi (y)$ simply as a function of $y$. We are for the moment concerned only with those values of $y$ which correspond to large positive values of $x$, that is to say with small positive values of $y$. And $\psi (y)$ has the property that by making $y$ sufficiently small we can make $\psi (y)$ differ by as little as we please from $l$. To put the matter more precisely, the statement expressed by $lim\varphi (x)=l$ means that, when any positive number $ϵ$, however small, is assigned, we can choose $x_0$ so that $|\varphi (x)–l|<ϵ$ for all values of $x$ greater than or equal to $x_0$. But this is the same thing as saying that we can choose $y_0=1/x_0$ so that $|\psi (y)–l|<ϵ$ for all positive values of $y$ less than or equal to $y_0$.

We are thus led to the following definitions:

A. If, when any positive number $ϵ$, however small, is assigned, we can choose $y_0(ϵ)$ so that $$|\varphi (y)–l|<ϵ$$ when $0<y\le y_0(ϵ)$, then we say that $\varphi (y)$ tends to the limit $l$ as $y$ tends to $0$ by positive values, and we write $$\underset{y\to +0}{lim}\varphi (y)=l.$$

B. If, when any number $\mathrm{\Delta }$, however large, is assigned, we can choose $y_0(\mathrm{\Delta })$ so that $$\varphi (y)>\mathrm{\Delta }$$ when $0<y\le y_0(\mathrm{\Delta })$, then we say that $\varphi (y)$ tends to $\mathrm{\infty }$ as $y$ tends to $0$ by positive values, and we write $$\varphi (y)\to \mathrm{\infty }.$$

We define in a similar way the meaning of ‘$\varphi (y)$ tends to the limit $l$ as $y$ tends to $0$ by negative values’, or ‘$lim\varphi (y)=l$ when $y\to -0$’. We have in fact only to alter $0<y\le y_0(ϵ)$ to $-y_0(ϵ)\le y<0$ in definition A. There is of course a corresponding analogue of definition B, and similar definitions in which $$\varphi (y)\to -\mathrm{\infty }$$ as $y\to +0$ or $y\to -0$.

If $\underset{y\to +0}{lim}\varphi (y)=l$ and $\underset{y\to -0}{lim}\varphi (y)=l$, we write simply $$\underset{y\to 0}{lim}\varphi (y)=l.$$ This case is so important that it is worth while to give a formal definition.

If, when any positive number $ϵ$, however small, is assigned, we can choose $y_0(ϵ)$ so that, for all values of $y$ different from zero but numerically less than or equal to $y_0(ϵ)$, $\varphi (y)$ differs from $l$ by less than $ϵ$, then we say that $\varphi (y)$ tends to the limit $l$ as $y$ tends to $0$, and write $$\underset{y\to 0}{lim}\varphi (y)=l.$$

So also, if $\varphi (y)\to \mathrm{\infty }$ as $y\to +0$ and also as $y\to -0$, we say that $\varphi (y)\to \mathrm{\infty }$ as $y\to 0$. We define in a similar manner the statement that $\varphi (y)\to -\mathrm{\infty }$ as $y\to 0$.

Finally, if $\varphi (y)$ does not tend to a limit, or to $\mathrm{\infty }$, or to $-\mathrm{\infty }$, as $y\to +0$, we say that $\varphi (y)$ oscillates as $y\to +0$, finitely or infinitely as the case may be; and we define oscillation as $y\to -0$ in a similar manner.

The preceding definitions have been stated in terms of a variable denoted by $y$: what letter is used is of course immaterial, and we may suppose $x$ written instead of $y$ throughout them.

94. Limits as $x$ tends to $a$.

Suppose that $\varphi (y)\to l$ as $y\to 0$, and write $$y=x–a,\varphi (y)=\varphi (x–a)=\psi (x).$$ If $y\to 0$ then $x\to a$ and $\psi (x)\to l$, and we are naturally led to write $$\underset{x\to a}{lim}\psi (x)=l,$$ or simply $lim\psi (x)=l$ or $\psi (x)\to l$, and to say that $\psi (x)$ tends to the limit $l$ as $x$ tends to $a$. The meaning of this equation may be formally and directly defined as follows:

if, given $ϵ$, we can always determine $\delta (ϵ)$ so that $$|\varphi (x)–l|<ϵ$$ when $0<|x–a|\le \delta (ϵ)$, then $$\underset{x\to a}{lim}\varphi (x)=l.$$

By restricting ourselves to values of $x$ greater than $a$, i.e. by replacing $0<|x–a|\le \delta (ϵ)$ by $a<x\le a+\delta (ϵ)$, we define ‘$\varphi (x)$ tends to $l$ when $x$ approaches $a$ from the right’, which we may write as $$\underset{x\to a+0}{lim}\varphi (x)=l.$$ In the same way we can define the meaning of $$\underset{x\to a-0}{lim}\varphi (x)=l.$$ Thus $\underset{x\to a}{lim}\varphi (x)=l$ is equivalent to the two assertions $$\underset{x\to a+0}{lim}\varphi (x)=l,\underset{x\to a-0}{lim}\varphi (x)=l.$$

We can give similar definitions referring to the cases in which $\varphi (x)\to \mathrm{\infty }$ or $\varphi (x)\to -\mathrm{\infty }$ as $x\to a$ through values greater or less than $a$; but it is probably unnecessary to dwell further on these definitions, since they are exactly similar to those stated above in the special case when $a=0$, and we can always discuss the behaviour of $\varphi (x)$ as $x\to a$ by putting $x–a=y$ and supposing that $y\to 0$.

95. Steadily increasing or decreasing functions.

If there is a number $\delta$ such that $\varphi (x^{\prime })\le \varphi (x”)$ whenever $a–\delta <x^{\prime }<x”<a+\delta$, then $\varphi (x)$ will be said to increase steadily in the neighbourhood of $x=a$.

Suppose first that $x<a$, and put $y=1/(a–x)$. Then $y\to \mathrm{\infty }$ as $x\to a-0$, and $\varphi (x)=\psi (y)$ is a steadily increasing function of $y$, never greater than $\varphi (a)$. It follows from § 92 that $\varphi (x)$ tends to a limit not greater than $\varphi (a)$. We shall write $$\underset{x\to a+0}{lim}\varphi (x)=\varphi (a+0).$$ We can define $\varphi (a-0)$ in a similar manner; and it is clear that $$\varphi (a-0)\le \varphi (a)\le \varphi (a+0).$$ It is obvious that similar considerations may be applied to decreasing functions.

If $\varphi (x^{\prime })<\varphi (x”)$, the possibility of equality being excluded, whenever $a–\delta <x^{\prime }<x”<a+\delta$, then $\varphi (x)$ will be said to be steadily increasing in the stricter sense.

96. Limits of indetermination and the principle of convergence.

All of the argument of §§ 80-84 may be applied to functions of a continuous variable $x$ which tends to a limit $a$. In particular, if $\varphi (x)$ is bounded in an interval including $a$ (i.e. if we can find $\delta$, $H$, and $K$ so that $H<\varphi (x)<K$ when $a–\delta \le x\le a+\delta$).¹ then we can define $\lambda$ and $\mathrm{\Lambda }$, the lower and upper limits of indetermination of $\varphi (x)$ as $x\to a$, and prove that the necessary and sufficient condition that $\varphi (x)\to l$ as $x\to a$ is that $\lambda =\mathrm{\Lambda }=l$. We can also establish the analogue of the principle of convergence, prove that the necessary and sufficient condition that $\varphi (x)$ should tend to a limit as $x\to a$ is that, when $ϵ$ is given, we can choose $\delta (ϵ)$ so that $|\varphi (x_2)–\varphi (x_1)|<ϵ$ when $0<|x_2–a|<|x_1–a|\le \delta (ϵ)$.

97.

The reader will probably fail to see at first that any proof of such results as those of Exs. 4, 5, 6, 7, 8 above is necessary. He may ask ‘why not simply put $x=0$, or $x=a$? Of course we then get $a$, $a/\alpha$, $a^m$, $P(a)$, $R(a)$’. It is very important that he should see exactly where he is wrong. We shall therefore consider this point carefully before passing on to any further examples.

The statement $$\underset{x\to 0}{lim}\varphi (x)=l$$ is a statement about the values of $\varphi (x)$ when $x$ has any value distinct from but differing by little from zero.² It is not a statement about the value of $\varphi (x)$ when $x=0$. When we make the statement we assert that, when $x$ is nearly equal to zero, $\varphi (x)$ is nearly equal to $l$. We assert nothing whatever about what happens when $x$ is actually equal to $0$. So far as we know, $\varphi (x)$ may not be defined at all for $x=0$; or it may have some value other than $l$. For example, consider the function defined for all values of $x$ by the equation $\varphi (x)=0$. It is obvious that $$\begin{array}{}\text{(1)}& lim\varphi (x)=0.\end{array}$$ Now consider the function $\psi (x)$ which differs from $\varphi (x)$ only in that $\psi (x)=1$ when $x=0$. Then $$\begin{array}{}\text{(2)}& lim\psi (x)=0,\end{array}$$ for, when $x$ is nearly equal to zero, $\psi (x)$ is not only nearly but exactly equal to zero. But $\psi (0)=1$. The graph of this function consists of the axis of $x$, with the point $x=0$ left out, and one isolated point, viz. the point $(0,1)$. The equation expresses the fact that if we move along the graph towards the axis of $y$, from either side, then the ordinate of the curve, being always equal to zero, tends to the limit zero. This fact is in no way affected by the position of the isolated point $(0,1)$.

The reader may object to this example on the score of artificiality: but it is easy to write down simple formulae representing functions which behave precisely like this near $x=0$. One is $$\psi (x)=[1–x^2],$$ where $[1–x^2]$ denotes as usual the greatest integer not greater than $1–x^2$. For if $x=0$ then $\psi (x)=[1]=1$; while if $0<x<1$, or $-1<x<0$, then $0<1–x^2<1$ and so $\psi (x)=[1–x^2]=0$.

Or again, let us consider the function $$y=x/x$$ already discussed in Ch. II, § 24, (2). This function is equal to $1$ for all values of $x$ save $x=0$. It is not equal to $1$ when $x=0$: it is in fact not defined at all for $x=0$. For when we say that $\varphi (x)$ is defined for $x=0$ we mean (as we explained in Ch. II, l.c.) that we can calculate its value for $x=0$ by putting $x=0$ in the actual expression of $\varphi (x)$. In this case we cannot. When we put $x=0$ in $\varphi (x)$ we obtain $0/0$, which is a meaningless expression. The reader may object ‘divide numerator and denominator by $x$’. But he must admit that when $x=0$ this is impossible. Thus $y=x/x$ is a function which differs from $y=1$ solely in that it is not defined for $x=0$. None the less $$lim(x/x)=1,$$ for $x/x$ is equal to $1$ so long as $x$ differs from zero, however small the difference may be.

Similarly $\varphi (x)=\{(x+1{)}^2–1\}/x=x+2$ so long as $x$ is not equal to zero, but is undefined when $x=0$. None the less $lim\varphi (x)=2$.

On the other hand there is of course nothing to prevent the limit of $\varphi (x)$ as $x$ tends to zero from being equal to $\varphi (0)$, the value of $\varphi (x)$ for $x=0$. Thus if $\varphi (x)=x$ then $\varphi (0)=0$ and $lim\varphi (x)=0$. This is in fact, from a practical point of view, from the point of view of what most frequently occurs in applications, the ordinary case.