XV

proceed to give Weierstrass's explanation of its real meaning.

In the first place notice that, in discussing $2x+h$, we have been considering $x$ as fixed in value and $h$ as varying. In other words $x$ has been treated as a "constant" variable, or parameter, as explained in IX.; and we have really been considering $2x+h$ as a function of the argument $h$. Hence we can generalize the question on hand, and ask what we mean by saying that the function $f(h)$ tends to the limit $l$, say, as its argument $h$ tends to the value zero. But again we shall see that the special value zero for the argument does not belong to the essence of the subject; and again we generalize still further, and ask, what we mean by saying that the function $f(h)$ tends to the limit $l$ as $h$ tends to the value $a$.

Now, according to the Weierstrassian explanation the whole idea of $h$ tending to the value $a$, though it gives a sort of metaphorical picture of what we are driving at, is really off the point entirely. Indeed it is fairly obvious

that, as long as we retain anything like "$h$ tending to $a$," as a fundamental idea, we are really in the clutches of the infinitely small; for we imply the notion of $h$ being infinitely near to $a$. This is just what we want to get rid of.

Accordingly, we shall yet again restate our phrase to be explained, and ask what we mean by saying that the limit of the function $f(h)$ at $a$ is $l$.

The limit of $f(h)$ at $a$ is a property of the

neighbourhood of $a$, where "neighbourhood" is used in the sense defined in XI. during the discussion of the continuity of functions. The value of the function $f(h)$ at $a$ is $f(a)$; but the limit is distinct in idea from the value, and may be different from it, and may exist when the value has not been defined. We shall also use the term "standard of approximation" in the sense in which it is defined in XI. In fact, in the definition of "continuity" given towards the end of that chapter we have practically defined a limit. The definition of a limit is:–-

A function $f(x)$ has the limit $l$ at a value $a$ of its argument $x$, when in the neighbourhood of $a$ its values approximate to $l$ within every standard of approximation.

Compare this definition with that already given for continuity, namely:–-

A function $f(x)$ is continuous at a value $a$ of its argument, when in the neighbourhood of $a$ its values approximate to its value at $a$ within every standard of approximation.

It is at once evident that a function is continuous at a when (i) it possesses a limit at a , and (ii) that limit is equal to its value at a . Thus the illustrations of continuity which have been given at the end of XI. are illustrations of the idea of a limit, namely, they were all directed to proving that f ( a ) was the limit of f ( x ) at a for the functions considered and the value of a considered. It is really more instructive to consider the limit at a point where a function is not continuous. For example, consider the function of which the graph is given in [fig.]20 of XI. This function f ( x ) is defined to have the value 1 for all values