You are reading a page from An Elementary Treatise on Actuarial Mathematics by Harry Freeman (1932)
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CHAPTER IX

FUNCTIONS AND *LIMITS
FUNCTIONS. RATES    135
the other hand, if the relation connecting x and y is of the form ck (x, y) = a + bxy + cx2y + dy3 = o, we cannot find y by an immediate substitution of a value of x. A further process is necessary—in this example the solution of a cubic equation in y—before the value or values of y can be obtained. q (x, y) = o defines an implicit function of x and y. It should be noted that plane curves can be represented either by an explicit function of one variable,
y = f (x) ; or by an implicit function of two variables defined by 6 (x, y) = o. Similarly, an explicit function of two variables,
z = f (x, y) and an implicit function of three variables defined by
(x, y, z) = o represent surfaces in three-dimensional geometry.
A familiar type of rational integral function to which reference has already been made (Chap. p. 6o) is a homogeneous function.
f (x, y, z, ...) is a homogeneous function of the nth degree in x, y, z, ... if, when the variables x, y, z, ... are replaced by Ax, Ay, Az, ... respectively, the resulting function is An f (x, y, z, ...).
A simple example is
L (x3 + y3 + z3) +M(x2y+y2z+z2x+xy2+yz3+zx2)
+ Nxyz;
this is a homogeneous function of the third degree in x, y and z.
136    FUNCTIONS AND LIMITS
interval, whatever the values of a, b, c, d, ..., Ay/Ax is constant, then y is said to vary at a constant rate with respect to x.
It is evident that this constant variation will occur only in a limited number of instances. A well-known example is that of uniform motion in a straight line. If x represents time-intervals and y distance-intervals, the ratio Ay/Ax represents the speed of the moving body, and if this ratio is constant, the body is said to be moving uniformly or at a constant rate.
More commonly, rates will be variable and the successive values of 60'/Ox will not be equal. We can, however, assign a meaning to Ay/6,x by considering each interval separately. For example, giving numerical values to x and y, uniform variation is illustrated by
x    1 2 3 4
Y    5 10 15 20
for Ay/Ax = 5 = constant.
On the other hand, if corresponding values of x and y are
2 3 4 12 30 6o
.1y/Ax takes the values 7, 18, 30, ... for successive intervals, and is variable. If, however, we were to consider the range 1 to 4 for x, we could say that over this range of values of x, y increases from 5 to 6o, and that the average rate of increase of y over this range = (6o — 5)/(4 — I) = 55/3.
We are led therefore to the following definition :
Given corresponding increments h and k in the values of x and y for the function y = f (x), the average rate of variation of y with x is the uniform rate which would give an increment k in the value of y for the increment h in x.
5. The average rate of variation over an interval has been illusabove by a body moving with variable speed. This is the speed over an interval of time, and its meaning can easily be appreAnother conception of the term " speed" is that of speed at a particular moment of time. Suppose that the distance travelled by a moving body varies with the square of the time that has elapsed

    
RATES    '37
since the beginning of the motion, so that s = t2. The average
speed over an interval At will be
(t+ot)2—t2 (t+At)—t    or
Giving At the values i, •1, •oi, •ooi, ... we may construct the following table :
Interval    ttot+I ttot+•I ttot+.oi ttot+•ool...
Average speed
over interval 2t + I    zt + .1    2t + •oI    2t + '00I ...
Now the average speed over an interval tends to become more nearly equal to the speed at the beginning of the interval as the interval is reduced. The average speed over the interval tends to the value at, and this must therefore be the value of the speed at the beginning of the interval.
More generally, the average rate of variation over an interval tends to the rate of change at a particular point (the beginning of the interval) as the interval is reduced.
It should be noted that although the value of
(t+zt)2—t2 (t + fit) — t
tends to 2t as At is reduced, we cannot put At = o at once, for we then obtain which is meaningless in algebra. (This is what might be expected, for the average speed over a non-existent interval has no meaning.)
Suppose now that for the function y = f (x) we take two successive values of the argument, namely, x and x + h. Then the average rate of variation of f (x) in the interval x to x + h will be
f (x + h) — f (x) f(x+h)—f(x)
(x+h)—x    h
which tends to the rate of change off (x) at the point x as h is reduced. This rate of change is therefore the limiting value of the average rate of change as h tends to zero, and we must reach this limiting value by a process other than by direct substitution of h = o in the algebraic expression.

138    FUNCTIONS AND LIMITS
6. Certain limiting values may be illustrated by the application of the methods of elementary geometry.
Example 1.
Let A be a fixed point on a plane curve and let B1AC1 be any straight line drawn through A cutting the curve again at B1. Let Bl move down the curve towards A so that the secant takes up the successive positions
C C3 C2 C,
Fig. 14.
B2AC2, B3AC3, .... Then the lengths of the secants cut off by the curve, namely B1 A, B2A, B3A, ... , become successively smaller. When, how-ever, the two points BA virtually coincide, the secant approaches the position B„AC,,, the tangent to the curve at the point A. In other words, the tangent BnACn is the limiting position of the secant B,AC, as B moves along the curve to A.
Example 2.
Prove that sin 0 < B < tan B.
Let KOA be the angle B (< a ).
Draw a circle with OA as radius and let AB be a chord of the circle. Draw AT, BT, the tangents to the circle at A and B respectively, meeting at T. Then evidently the chord AB<arcAB<AT+ TB; i.e. HA < arc AK < AT.
HA arc AK_ AT OA < radius OA < OA ' or    sinO<B<tanB.

Fig. 15.

CONTINUOUS FUNCTIONS From these inequalities we have
B    I
sin B c cos B' sin 0
i.e.    I > —B > cos B.
Therefore sin 6 lies between I and cos B. In the limiting case when B is zero, cos 0 is 1. (See Chapter I, para. I1.)
Therefore when 0 approaches the limit zero, sle 9 has I as its limiting value.
7. Continuous functions.
Before proceeding further to the consideration of limits and limiting values it is necessary to distinguish between those functions which vary continuously between two values of the argument and those which do not.
If we wished to plot the curve of the function y = x2 for all real values of x, we could give x certain values, and by substituting these values in the equationy = x2 we could obtain the corresponding values of y. It would be necessary to plot only a limited number of points (x, y) and by drawing a smooth curve through these points the graph of the function y = x2 would result. Suppose, however, that a limitation were imposed upon the values of x, namely, that x
Fig. 16.
should always be a positive integer. The graphical representation of the values of x and y would be a series of isolated points, and a curve could not be drawn between any two successive values of
(x, y).

140    FUNCTIONS AND LIMITS
Again, consider the function y2 = (x — i) (x — 2) (x — 3). If y is to be real we have the following conditions (I) x must not be less than I ; (2) x must not lie between the values x = 2 and x = 3. This second condition shows that while x may have any value between z and 2 and any value greater than 3, for real values of y, there is no value of y corresponding to values of x between z and 3. y is said to be discontinuous between the values x = 2 and x = 3, and the curve will take the above shape (Fig. i6).
A type of function which, for a certain value of the variable, ceases to be continuous is y = 1/x. If x be zero, the function takes the form I/o which is, strictly speaking, meaningless. As, however, 1/x becomes successively greater on decreasing x, it is possible to make I/x greater than any finite value, by making x sufficiently small. The function is then said to " tend to infinity " or to " increase indefinitely" as x tends to zero.
8. Limits.
We are now in a position to give a clearer definition of what is meant by a limit. A simple definition is as follows :
If y = f (x) and y tends continuously towards a certain value 1, and can be made to differ as little as we please from that value by making x approach some fixed value a, then 1 is said to be the limiting value of f (x) as x tends to the value a.
This may be expressed shortly as
Lt f (x) = 1. x-4-a
For example, we have
Lt f (x + h) — f (x) = 2X when f (x) = x2.
h+0    h
This definition is not sufficiently precise, and may prove in-adequate in certain instances. Consider for example the following illustration.
sin x
The curve y =    i x      is represented geometrically (see Fig. 17).


For large values of x the curve becomes indistinguishable from the axis of x and the value of y tends to zero, notwithstanding that, however large x may be, y may be sometimes increasing numeric-ally. It is obvious that the phrase "tends continuously towards a

LIMITS    141
certain value 1" does not mean "constantly increases (or decreases) to the value 1."
Fig. 17.
Now take the curve y = sin x.
Here y does not tend to a limit as x becomes indefinitely great. It might be claimed, however, that y (i.e. sin x) tends to unity for sufficiently large values of x. If this were countered by the arguthat for a very large value of x, sin x was, say, the reply might be that the value of x was not sufficiently large, and that by
A v    Iv IL ,w
Fig. 18.


taking a larger value of x, sin x would differ from unity by as little as we pleased. The rejoinder to this would be that by taking a still larger value of x, sin x could be made to differ from 2, or zero, or -1-, etc. by as little as we pleased; and so on.


By the statement that Lt sin x is zero and that sin x can be x,. x    x
made to differ from zero by as little as we please, we imply that, given a number, say, •oi, we must be able to find a value of x such that, for all greater values of x, si x will differ from zero by less
n
than •oi. In other words, the whole of the graph of y = si x after


this point will be contained within the two ordinates y = •oi and y = — •01. Similarly, if the number •ooi were chosen, a value of

142    FUNCTIONS AND LIMITS
x must be found such that for all greater values of x, the graph will be contained in the limits y = •oor and y = — •ool.
It is clear that the graph of y = sin x would satisfy such a series x
of tests, but that the graph of y = sin x would not.
This leads directly to the more rigorous definition of a limit:
Let f (x) be a function such that x lies between two fixed values a and b (i.e. a < x < b) and let x' be any value of x satisfying these conditions. Then if I be a number such that corresponding to an arbitrary positive number E, a positive number 77 can be found such that f (x) differs from l by less than E whenever x — x' < 17, then 1 is said to be the limit of f (x) as x -÷ x'.
It should be emphasized that the limit off (x) as x a is not defined as a value off (x), and in particular is not necessarily equal to f (a). It is a quantity quite distinct from the values of f (x) although it is defined by means of these values in the neighbour-hood of x = a. As a rule, the limit off (x) as x — a is required in circumstances in which f (a) has no meaning.
9. It is a simple matter to prove that the algebraic sum, product or quotient of the limits of any finite number of functions is the limit of the sum, product or quotient respectively of the functions, provided that, when considering quotients, the limit of the divisor is not zero. The definition of a limit and these corollaries form the basis of the infinitesimal calculus.
The following elementary examples are typical of the methods employed in the evaluation of limits.
Example 3.
Find    Lt    
x3—a3
,a x — a
We may not put x = a immediately, for in that event the divisor will be zero and we shall arrive at the form ° . If we divide throughout by x — a the function becomes x= + ax + a2 and if we let x -+ a in this
— 3
expression we obtain 3a2. This is the limit when x a of      a x — a
Although it should be proved that a positive number 77 can be found
x3 — a3
such that         — 3a2 is less than any arbitrary number E whenever
x — a

    
ILLUSTRATIVE EXAMPLES    143 x — a < 77, it may be taken for granted that this criterion holds in all the examples that will be dealt with subsequently, and that we may proceed straight to the limit as above.


Example 4.
Find Lt 4 En3J where n is a positive integer. n-->o n    J
n2(n + I)2 n2 + 2n3 + n4
    In3= 13+23+33+... +n3=    =
4    4
n4En3=4[.2+n+ I]=4n2+2n+4.
Lt 1 En3 = Lt I + Lt I + Lt I
    n-*oon4    n->co4n2    n~m2n n->co 4
by the proposition above =0+0+4
Example 5.
Show that    Lt x5 — 5x + 4 = r.
X- Ix3—2x+I
If we put x = I immediately we obtain the form   As in Example 3, we could divide numerator and denominator by x — 1 and then find the limit. An alternative method is as follows :
Put x = 1 + h ; then the function becomes a function of h instead of a function of x and we have to find the limit of the new function when h --~- o.
x6 —Sx+4—(I+h)s— 5(1+h)+4 x3 -2x + I    (1 + h)3 — 2 (1 + h) +1
_I+6h+15h2+...—5—5h+4 1+3h+3h2+h3—2—2h+I
_h+ 15h2+... h+3h' +...
I + 15h + ... I + 3h + ...
and the limit of this expression when h , o is I. 10. Limit of a sequence.
Let uI + u2 + u3 + u4 + ... be an infinite series. If s,, be the sum of the first n terms, we can form an unending sequence of

144    FUNCTIONS AND LIMITS
values, sl, s2, s3 ... sn .... If un    o as n -+ oo s„ may tend to a finite number. For example, the( series
I + + (2)2 + (2)3 + ... + (2)n-1 = [I — (2M(1 — 2) = 2 —
so that sn, the sum to n terms, differs from 2 by the small quantity
(2 )n-1.
The larger the value of n, the more nearly the sum to n terms is equal to 2; the sum 2 as n -~ co. A series of a different type is
I+2+3+4+...+n.
The sum to n terms is 2n (n + I), and there is no fixed number to which the sum of the series tends: if n be very large In (n + I) is very large.
In the first example the limit of the sequence as n increases in-definitely is said to be 2, and the series is said to be convergent. In the second example there is no limit to the sum of the series, and the series is said to be divergent.
The definition of the limit of a sequence is as follows :
If u1i u2, u3, ... un, ... be an unending sequence of real or imaginary numbers, and if a number 1 exists such that corresponding to every positive number (however small) a number k can be found such that un differs from l by less than e for all values of n > k, the
sequence u1, u2 i u3, ... un, ... is said to tend to the limit l as n -> al).
The limits of algebraic and other expansions are of the utmost importance in mathematical work, and while it is beyond the present scope to examine fully the convergence or otherwise of even the more important series, reference to them is essential for the proper understanding of the calculus.
11. (x+h)n—xn
h
If n be a positive integer, this expression becomes

h [n1hxn-1 + n2h2xn-2 + n3h3xn-3 + ... + hn]
= nlxn-1 + n2hxn-2 + n3h2xn-3 + ... + 1n-1 which evidently tends to nxn-1 when h —* o.

LIMITS    145
Suppose, however, that n be other than a positive integer. Then there will not be a limited number of terms, and we have
Lt (x + h)"" — xn _ Lt xn (1 +    — 1 h-->0
h    h-->O
h    l
= hLto h Cn1(x)+n2(x)2+n3(xl3+...~ .
This involves a double limit, for the number of terms inside the bracket is not finite, and we are not entitled to assume that the limit of the sum of the terms is equal to the sum of the limits of the terms.
The investigation of a double limit requires further mathematical analysis, and the consideration of the limit of the above expression when n is not a positive integer will be deferred to a later chapter.
For all values of r it may be shown that
lies between
The expression
is denoted by e, so that
where    R <
r.r!'
Since, however,    Lt
r,o r.r!
is zero, e may be considered as the sum of the infinite series
+    2 (+ 1
+ ... + - 1 - + ....

146    FUNCTIONS AND LIMITS Again it may be shown that
/1 + x1'' > + x x2 2 x3
    Lt (    xr
    /    I+ +—•••+xr
    n,w \    n    2! 3!    Y!~
    2    r    r+l
but    <1+x+2!+3! +•••+Y~+(Y+—x)r!'
and that ex= Lt (I+x)n=i+x+x2+x3+...+xr+
n—>co    n    2!    3!    r!
if x is not zero.
In the inequalities above put r =
Then    ex > 1 + x and < 1 + x +
(2—x)'
ex—1>x and <x+ x2
ex— I    x
i.e.    > I and < 1 +
x    2—x'
so that if x be positive Lt ex -- 1 =1.
x—o x
If x is negative we can replace x by — y and obtain .
    ex—    e-y_1=    I ey — ILt    = Lt    Lt         =1.
    x—o x    v-*o —y    v_oe'' y


13. We may now proceed to a more formal definition of a confunction.
ASYMPTOTES 147 between — 1 and + 1, namely where x = o, y takes the value 1!o, which is not a finite number. The function is therefore discontinuous at the point x = o (cf. para. 7 above).


14. Asymptotes.
x2 Consider the curve    y = (x _     1)2•
Here y tends to infinity as x    1, and since we may write the
equation as y = (1 1      42, y tends to the value 1 when x tends to
infinity in either direction.
The curve is of the following shape.
Fig. iq.
Discontinuities in the value of y when x = 1 and of x when y = 1 are apparent. It will be seen that the curve gradually approaches indefinitely near to the straight lines x = 1 and y = 1 but does not actually meet them at any finite distance from the origin.
Such lines are called asymptotes to the curve.
Example 6.
Find the asymptotes to the curve y = ~3x    — I) (x — 2)
(x—3)(x+3)The equation of the curve may be written
    29 —    7x    4    25
3 (x—3)(x+3)3    3(x—3) 3(x--3)
Then if x tend to infinity in either direction, the curve approaches the straight line y = 3. Hence y = 3 is an asymptote.
148    FUNCTIONS AND LIMITS
Further, if x is positive and greater than 29'7, the value of y is less than 3. Therefore, on the right the curve approaches y = 3 from underIf x is negative, on the left the curve approaches y = 3 from above.
Again, from the second form of the equation to the curve, it will be seen that x = 3 and x = — 3 are asymptotes to the curve, since the curve gradually approaches these straight lines but does not meet them at a finite distance from the origin.
x,
x
Y,    1
Fig. 20.
EXAMPLES 9
x9—ap
r. Find Lt x4   —     aq where p and q are positive integers.
x-+¢
Y2
s and hence show that the limit of the sum when n
r=n
EXAMPLES
1/x4 + axe + bx + c — 1/x4 + kx2 + snx + n
when x is indefinitely increased.
 3.5.7 • • • 2n+ I > 4.6.8 ... 2n + 2. 2.4.6    2n    3.5.7 ... 2n+ I'

19. From the inequalities in Question 18 prove that Lt 1_3.5...2n—I=
    n_>.ao    2.4.6 ... 2n

Lt 3.5.7...2n+I—. 2.4.6 ... 2n
Prove also that the limit when n — co of the product of these two funclies between and I.
rr    I ln-1