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

6

IJLUSORY FUNCTIONS.

Ex. L. 2 - X
A fraction Thus, if f(m) = 2-, the operations represented by f
which as- -
sumes the
o cease to give any definite result when x = 1; for then f(x)
form
orm - - 1-I 0
assumes the form, or-, which is not a definite result,
1 —1 0'
as is shewn in the note*; f(X) therefore becomes illusory when
-= 1.
Ex.2. 15. Again, iff (x) = (1 + x)-, the operations represented
(l+-x) by f cease to give a definite result when x = 0; for then
when x=0. = i
(1 + x); assumes the form 1i, which is not a definite result,
as is shewn in the notet; f(x) therefore becomes illusory, in
this case, when x = o.
Ex. 3.fo 16. Again, if ABC (fig. 4) be an ellipse marked with the
Case of two i
intersecting usual letters, PG the line bisecting the angle SPH, and
0 0 0
5does not *That 0-is not a definite quantity appears thus. ~- according to the strict
represent definition of a quotient, is that quantity which multiplied by 0 gives 0: now any
any definite 0
quantity. quantity whatever multiplied by 0 gives 0; therefore 0 is any quantity whatever;
i. e. it is not a definite quantity.
It may be said, however, that though 0, considered absolutely, is not a definite
quantity, nevertheless — 2 becomes 2 when = 1; for x_ =,+ and _
X2 - 1 2 x" - 1 - x ~X~ 1 ' x + 1 -, 2
when = 1, and therefore _ = when = 1.
To this we may answer, that 2 _ is proved to be equal to + i by dividing its
numerator and denominator by x-1; but we may not perform this division when
x- 1 = 0, since there is no rule of Algebra which enables us to divide the numerator
and denominator of a fraction by zero without altering its value; hence the equation
2 X
2 = holds only on the express condition that x does not = 1; and therefore
we may not draw any conclusion from this equation which requires us to suppose
X2 __
that x actually = 1: consequently, we cannot assert that 2- 1 - when = 1 be
cause x2_=-+
1 1 1
16 not a t That 1~ is not a definite quantity may be shewn thus. 1~ is that quantity
definite which taken to the power 0 becomes 1; now any quantity whatever taken to the
quantity. I
power 0 becomes I; therefore 1~ is not a definite quantity.