An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

LEMMAS RESPECTING LIMITING VALUES.

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proaches a, f (x) has the same sign as A for all values of x
taken sufficiently near a.
Since f(x) - A may be diminished ad libitum by taking
x near enough to a, it is clear that f(x) may be made
greater or less than zero, according as A is greater or less
than zero; or in other words, f(x) may be made to have
the same sign as A by taking x near enough to a: and when
x is made to approach still nearer to a, since we so diminish
f(a) - A still more, at least for all values of x near enough
to a (see 31), f (v) must continue to have the same sign as A.
Hence f(x) has the same sign as A for all values of x taken
near enough to a. Q.E.D.
37. If f (x) be any function of x which becomes illusory Lemma V.
when x = a certain value a, and if for each value of x (a
of course excepted) f (x) has only one value; then f (x)
cannot have more than one limiting value when x approaches a.
If possible let two different quantities A and B be both
limiting values off(v) when x approaches a; then we may
make f (x) differ from both A and B as little as we please,
at the same time, by sufficiently diminishing - a; therefore x may be so taken, that A and B shall differ from the
same quantity f (x) since f () has only one value}, and
therefore from each other, as little as we please; which is
absurd if A and B be two different quantities; therefore A
must be equal to B. Hence there cannot be more than one
limiting value. Q. E.D.
38. Hence if we can prove that A is a limiting value of
f(I) when x approaches a, we are sure that no other quantity
but A is a limiting value, and therefore that A is the limiting
value.
Hence it appears that a limiting value is not a mere Alimiting
approximation, but a perfectly definite quantity; for if it value isnt
a mere apwere a mere approximation, then, when we had found a limit- proximate
ing value A, any quantity differing very little from A would q
be just as much a limiting value as A; contrary to what has
been just proved.
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