The Calculus Primer (2011)
Part I. Functions, Rates, and Limits
Chapter 3. THE LIMIT CONCEPT
1—11. The Idea of a Limit Is Familiar. The reader may not realize that he is already familiar with the notion of a limiting value from his geometry. For example, he will recall that the circumference of a circle may be regarded as the limit of the perimeters of a series of inscribed and circumscribed regular polygons as the number of sides becomes indefinitely greater. This simply means that as the number of sides becomes infinitely greater, the differencebetween the length of the perimeter and the length of the circle becomes smaller and smaller; it does not mean that eventually the length of the circle equals the perimeter of any particular polygon, or that a polygon can ever coincidewith a circle.
Again, from his algebra, the reader will recall that the sum of an infinite number of terms of a geometric series, when r < 1, also approaches a limit, namely, the value . For example, in the series to infinity, it will be recalled that as more and more terms are taken, the sum in each instance becomes greater. It appears, also, that as the number of terms increases, the sum, while getting larger, is increasing less rapidly; in other words, there appears to be an upper limit to the sum, beyond which we can never go, no matter how many terms are taken—even an infinite number. In the case of the present series, that limit equals 2, as may be seen from the geometric representation of the terms of the above series by means of the segments on a line 2 units long, formed by successively bisecting the right-hand halves. At least intuitively it is clear that even when we have added all the terms that can possibly be thought or imagined—the series never ends, however—the sum could never exceed 2. As long as we add a finite number of terms, the sum will always be a little less than 2; but we can make the sum as close to 2 as we wish, merely by taking a sufficiently large number of terms.
1—12. Sharpening Our Ideas. When we think of the speed of a moving object at a particular instant, exactly what do we mean? We certainly do not mean the average speed for the next hour, or half hour, or minute, or even the next second. But we have just seen (§1—10) that the average speed for a very small interval is a close approximation of the instantaneous speed we have in mind. By taking the interval still smaller, we can make the approximation still closer; in fact, as close as we wish, if the interval is taken sufficiently small.
In other words, we may think of the instantaneous speed at a certain instant as the limiting value which the average speed would approach if the interval were indefinitely shortened, while always including the instant in question.
The distinction to be appreciated is that an interval of time has extent, while an instant of time has none. No distance, however small, can be covered “during an instant”; indeed, the phrase is meaningless, since an instant has no duration. Thus we cannot say an instantaneous rate is “the distance covered during the instant divided by the duration of the instant.” Furthermore, it is equally meaningless to refer to the “speed during the shortest possible interval of time”; this is merely an average rate once more, since any “possible interval,” however small, has some size or extent. Moreover, “shortest interval” can mean nothing, since any interval, however small, can always be subdivided into millions or billions, etc., of still smaller intervals. Finally, we must not allow ourselves to slip into the phraseology, the “rate at an instant.” This begs the question: there is no motion at an instant for even an exceedingly small interval.
1—13. Limit Defined. Is there, then, no precise way of defining an instantaneous rate? There is, by using the idea of a limiting value, already referred to in §1—11. We shall now define the term limit.
A variable v is said to approach a constant l as a limit when the successive values of v are such that the numerical value of the difference v − l ultimately becomes and remains less than any previously specified value, no matter how small.
As the reader comes to understand more fully the meaning of a limit, he will appreciate the fact that every word and phrase in the above definition is significant and essential. Indeed, an even more precise definition of the limiting value of a function will be presented later. It should be carefully noted that the question of whether a variable ever reaches its limit or not has nothing to do with the question of its approaching a limit. The crux of the matter is that the difference between v and l shall, sometime, as v varies, become less than, and thereafter remain less than, some specified number, however small.
1—14. Limiting Value of a Function. When dealing with limiting values, we use the notation v → l, which is read “v approaches l as a limit.” Consider the function f(x) = x^{2} + 3x + 1. When x = 2, f(x) = 11; when x = 4, f(x) = 29. As we take the value of x close to 3, the value of f(x) will be close to 3^{2} + 3(3) + 1, or 19. If we take values of x successively nearer to 3, so that |x − 3| becomes less and less, the quantity |f(x) − 19| will also become less and less. Now let us select any number, as small as we please, say . It is then possible to choose |x − 3| so as to make |f(x) − 19| less than . We may thus say that
x |
f(x) |
0 |
1 |
1 |
5 |
2 |
11 |
3 |
19 |
4 |
29 |
5 |
41 |
6 |
55 |
which is read “the limiting value of the function x^{2} + 3x + 1 as x approaches 3 is 19.” The limit in this illustration is exactly 19.
The limit of a function is the exact value toward which the function approaches as x comes sufficiently near to some fixed value. The value of the function may or may not reach the limit toward which it approaches. In general, the limit of f(x) as x approaches some constant a is equal to some other constant b; that is,
always provided, of course, that the value of f(x) can be made to differ from b by as little as we please whenever x is taken sufficiently near to a.
1—15. Laws of Limits. We shall set forth, without proof, certain basic principles concerning variables and limits. For a further discussion and proof of these theorems, the reader is referred to more advanced treatises on the calculus. For the purposes of an introductory study of the subject such as this, no harm will be done by foregoing mathematical rigor to some extent; in fact, there is much to be gained by so doing.
Let us consider that u, v, and w represent functions of an independent variable x. Let us also suppose that
Remember that a function is a dependent variable; thus, if u = f(x), then u is a variable. Then the following laws can be shown to hold.
I.The limit of the sum, or difference, of two variables is equal to the sum, or difference, of their respective limits.
Thus
II.The limit of the product of a constant and a variable equals the product of the constant and the limit of the variable.
Thus
III.The limit of the product of a finite number of variables is equal to the product of their respective limits.
Thus
IV.The limit of the quotient of two variables is equal to the quotient of their respective limits, provided that the limit of the denominator is not zero.
Thus
EXAMPLE 1.
EXAMPLE 2.
EXAMPLE 3.
EXAMPLE 4.
EXAMPLE 5.
1—16. Infinity and Zero. We shall have occasion to make frequent use of another basic idea in connection with limits. If the numerical value of a variable x ultimately becomes and remains greater than any specified number, no matter how great, we say that x becomes infinite. This does not mean that infinity ( ∞ ) is a number, or any fixed quantity. The symbol ∞ does not denote a constant. We do not say “x approaches infinity”; we say instead, “x becomes infinite.” This means that x increases without limit; we write
x → ∞.
If the values of x are positive only, we say that x becomes positively infinite, or x → + ∞ ; if the values of x are negative only, we say that x becomes negatively infinite, or x → − ∞. The symbol ∞ is thus not a symbol for any number, quantity, or value; used in conjunction with other symbols, it is part of an abbreviation for the limiting results of a process of change.
Understood in this way, we may employ the symbol of infinity to abbreviate two very important limits, namely:
These may be interpreted, informally, as follows. In equation [1], as the denominator x becomes smaller and smaller, the value of the quotient becomes greater and greater. Thus:
The quotient can literally be made as great as we wish; that is, no matter how great a number may previously be designated, the value of can be made greater than this number simply by selecting a sufficiently small value of x. But the indicated quotient has no “value”; there is no number which results from dividing 1 by 0. That is what is meant by saying that “division by zero is impossible.”
Similarly, in equation [2], as the denominator x becomes greater and greater, the value of the quotient becomes smaller and smaller. Thus:
The quotient can literally be made as small as we wish, or as close to zero as we like, simply by taking a value for x which is sufficiently great.
Other important and frequently used limits are given below for reference; in the symbolical equations [3] to [14], it should be noted that a > 0, and c ≠ 0.
EXAMPLE 1.
EXAMPLE 2.
EXAMPLE 3.
EXAMPLE 4.
EXAMPLE 5.
EXAMPLE 6.
EXAMPLE 7.
EXAMPLE 8.
EXAMPLE 9.
EXAMPLE 10.
Solution.Substituting 0 for x yields , which is indeterminate. Hence, divide both numerator and denominator by x^{2}, the lowest power of x in the function:
EXAMPLE 11.
Solution.Divide both numerator and denominator by x^{3}, the highest power of x in the function.
EXAMPLE 12.
Solution.
Hence
EXAMPLE 13.
Solution.Substituting x = 3 yields = which is indeterminate.
By factoring:
1—17. A More Precise Definition of a Limit. Suppose that a toolmaker were asked to fashion a piece of work to a specified “ideal” dimension D, with an allowable leeway or tolerance limit of ±.002 inch. Call the measured dimension of the finished piece D′ and the difference D′ − D = E. (In this discussion we need not consider the error of measurement, since its exclusion does not affect the argument.) By using appropriate tools and skill, he can make the value of E as close to zero as he wishes. He can therefore make E lie between −.002 and +.002 simply by making E sufficiently close to zero. How “close to zero” should this be? He can presumably find a positive number δ, which may be very small, such that if the work is properly executed, he will have
−δ < E < +δ(E ≠ 0).
We are now in a position to define the limiting value of a function more rigorously. Consider the function F(v), which is defined on the domain
v_{1} < v < kandk < v < v_{2}.
We then say that a number L is the limit of a function F(v) as v approaches the number k, or
if for every number > 0, however small, there is another number δ > 0 such that, when v is in the domain of definition of F,
0 < (v − k) < δ,
then|F(v) − L| < .
Note that when we say “F(v) approaches L,” we are also saying that F(v) − L approaches zero. Note also that it does not matter whether the function F(v) is defined at v = k or not.
EXAMPLE.For F(v) = v^{2}, k = 4,
To find a “tolerance” limit, || > 0, no matter how small, we ask: How close to k = 4 must v become in order to make values of F(v), or v^{2}, lie between (16 + ) and (16 − )? In short, we wish to find a number δ such that
|v^{2} − 16| < when 0 < |v − 4| < δ.
Since v^{2} − 16 = (v + 4) (v − 4), it is clear that we can make the factor (v − 4) small by taking v close to 4; if we do this, the other factor, (v + 4), becomes approximately equal to 4 + 4, or 8, when v is close to 4. Thus the latter factor, (v + 4), can be made less than 9, for example, if we say let |v − 4| < 1. If we do let |v − 4| < 1, then v must lie between 3 and 5, and hence |v + 4| < 9. We therefore have:
|v^{2} − 16| = |(v + 4)(v − 4)| < 9|v − 4|.
We can then put a further limitation upon v so that
9|v − 4| < , or |v − 4| <
So, if we finally take δ equal to or 1, whichever is smaller, we know that
|v^{2} − 16| < ,
provided that 0 < |v − 4| < δ. The diagram will clarify these ideas. Geometrically, F(v) = v^{2} lies between (16 + ) and (16 − ) when v lies between (4 + δ) and (4 − δ);