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

10

ACTUAL AND LIMITING VALUES.

Now e2w may be made to differ from e2a as little as we
please by making x approach nearer and nearer in magnitude
to a; therefore, as in the first example, e2a, and no other quantity but e2a, is the limiting value of f(x) when x approaches
a. Hence in this case when the actual value ceases to be a
definite quantity the limiting value does not.
AEx. 3 23. Lastly, let us consider the fourth example (see 17).
A line cutting a curve
in two We have seen that the operations here represented by 0
points
which be- cease to give a definite result when s = 0; i. e. the actual value
ncme co- of 0p(s) ceases to be a definite quantity when s= o. But not
incident.
so the limiting value; for let AM (= x) MP (= y) be the coordinates of the point P (fig. 6), AN(= x') NQ( y') the
co-ordinates of the point Q, draw OP parallel to MN, and let
y =f () be the equation to the curve: then we evidently
have
OQ Y'-y f(y ) -f(a)
tan PRX= _ = - =
OP x -x x -
and this equation is true, no matter how near x' may be to x,
provided x' be not actually equal to i, for then the triangle
f(') -f(v)
OPQ ceases to exist, and tan PRX and () - ) cease to
W0 -
have definite values.
Now for simplicity let us suppose the curve to be a parabola having the axis of y for its axis, and its equation being
IV2
accordingly y = -; then
f (') - f () 1 'V - co x' + t
x - - 4m oo - x 4m,
except when x' =.
Hence it is evident that, except when v' actually = -,
we have
5 (s) = z PRX = tan-'
s m o 4m
Now the second member of this equation may be made to