126 Mr. Boot on the Analysis of Discontinuous Functions. 
vanishing fraction under limiting circumstances, we cannot determine from its 
value under other circumstances. Probably it ceases to be a subject of value. 
But the limit to which the series of its known values approximates, is something 
which we may determine, and to which we are able to attach a definite meaning : 
and for all applications this is sufficient. In one form or other the conception of 
a limit is indispensable; but, justly considered, this is a new subject of thought, 
not the basis of a new condition of reasoning. 
in the integral calculus such general considerations may be applied to the 
case of those formule of definite integration, the evaluation of which is made to 
depend on the assumption that some quantity, &, which they involve, is positive. 
We may conceive of such formule as tending to limits, and of theorems in which 
the consideration of those limits is involved. Fourier’s theorem, as has been 
already remarked, is of this class. Our purpose will be accomplished if we shall 
establish this position, and prove its sufficiency by further illustrations. 
2 ; O—= x § 
2. If in the function tan , we suppose 7 < a,and / a positive quantity, 
k pp P q y 
then, as & is diminished, the limit of the values of the function will be 5 This 
is evident. 
If « =a, the limit is 0, the entire series of values being 0. 
If 2 >a, the limit is — 5 
Let 
Af(a) =f(a+ 4a) — f(a), (3) 
then 
at ie a+ Aa—w« ie Si 
Atan ra tan Sg tan ae (4) 
and applying what precedes to each term of the second member, we find that the 
ns . . 7 . . 
limit of their sum is 7, or gy oF 0, according as x lies between a and a + Aa, or 
is equal to a or a + Aa, or lies entirely without those limits. 
In what follows we shall suppose that & is thus diminished, so that by any 
expression involving & we shall understand the limit to which it approaches as / 
approaches to 0. Then 
! ————— tan) f(x) ==) (a) or oe or 0, (5) 
as 
