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.

when z = b, it is evident that by sufficiently diminishing z - b
we may diminish x - a ad libitum, and therefore f (t) - A, by
definition (30), and therefore (z) - A since 0 (z) = f(x);
therefore A is the limiting value of (p(z) when z approaches
b, by def. (30). Q.E.D.
Examples. Thus if in the function f(v) we put x = a +, and
therefore f (v) =-f a + = (z) suppose; then x = a when
z = co, and therefore the limiting value of f(x) when x approaches a is the same thing as that of P(z) or f (a +-)
when z approaches co.
e' — 1
Again, if f() = and we put e6= z, and therefore
- = log z, and therefore f(x) log (z) suppose; then
x evidently = 0 when z = 1, and therefore the limiting value
E - 1
of f(x) or when x approaches 0, and that of (p(z) or
Z —I
- when z approaches 1, are the same thing.
log xV
We shall find this method of transforming expressions
very useful in finding their limiting values.
LemmaIII. 35. Every Actual value is also a Limiting value.
We have defined f(a) to be the result of performing the
operations represented by f upon a; therefore, by the definition of an actual value in (19), f(a) represents the actual
of f(x) when x = a. But, from the assumption stated in
(29), and the definition of a limit in (30), it is evident that
f(a) is the limiting value of f(x) when x approaches a.
Hence the truth of the lemma is evident. Of course we here
suppose that f(a) is not illusory.
LemmaIV. 36. If A be the Limiting value of f(x) when x ap