We may illustrate this in the following manner : As
division is a repeated subtraction of equal quantities from
a given sum, suppose we have ten dollars : we say, " two
would go into ten five times 4" or, we could go five times
and take two dollars at each time. Now, how often
could we go and take nothing ? Evidently an infinite
number of times.
The truth of the proposition may be proved thus;

Divide :; — - ; we shall have



INTEODUCTOEY EEMAEKS. 105

Y^—=l-\-x+x^+x^+etc., to infinity;
and making x=l,

Q=l + 1 + 1 + 1+ etc., to infinity,

which is an infinite quantity.
a
•■0 •

8. Since the value of a fraction diminishes as the de-
nominator increases, it follows that, as the denominator
approaches in value toward infinity, the value of the frac-
tion approaches zero ; hence the limit of

a

X

is zero when x is equal to infinity.
9. As the denominator of a fraction becomes nearer
and nearer in value to the numerator, the fraction itself
approximates in value to unity ; and when the denomi-
nator becomes equal to the numerator, the value is unity.
That IS, —=^a°=l,

when «=m.
10. By the term limit, therefore, we mean that value
toward which a variable quantity is approaching, hut
which it never reaches so long as the number of terms is
finite.
We shall illustrate the foregoing by a few examples.
Ex. 1. Fin d^ the limit of the value of the fraction
2x+5



first, when x=0
When 03=0,

fraction is f.
Again, when

nominator by x.
3a;+6'
, and then when a5=a).
