the fraction {a}{x} becomes infinitely small ; that is, as the denominator of a fraction approaches to the value infinity, the fraction itself approaches to the value 0. This full verbal
statement is sometimes written
{a}{\infty} = 0.
183. Meaning of {0}{0}. The symbol {0}{0} may be indeterminate in form or in fact. Thus the value of {x^2-4}{x-2} when x = 2 is 0, but by putting the fraction in the form {(x +2)(x -2)}{x -2} we see that the expression is equivalent to x + 2, which becomes 4 when x = 2. Again, {x^3-a^3}{x - a} = {0}{0} when x = a, but by putting the fraction in the form {(x -a )(x^2+ax +a^2)}{x -a} we see that the expression is equivalent to x^2 + xa + a^2, or 3 }a^2, when x = a. These fractions assumed the form {0}{0} under particular conditions, but it is evident that they do not necessarily have the same value.
On the other hand, the symbol {0}{0} may show that a value is really indeterminate. Thus, solving in the regular way the equations
x+y + 2 = 0, 2x+2y+4=0,
we get x = {4-4}{2-2} = {0}{0}, and we can easily see that x can have any value whatever if we give y a value to suit, so that the value of x is indeterminate.
184. Meaning of {\infty}{\infty} Inasmuch as {1}{\infty} = 0, what is true of {0}{0} is equally so of {\infty}{\infty}
