134 Functions, limits, derivatives
psychologically satisfying. Whenever a positive number a is selected,
we can find a positive number N so large that f(x) approximates L so
closely that If(x) - Ll < e whenever x is so large that x > N. The
abbreviated version of this assertion is(3.341) lim f(x) = L.
X_WThis is read "the limit as x approaches infinity of eff of ex is ell," or "the
limit as x becomes infinite of eff of ex is ell." This does not mean that
"infinity" is a place toward which numbers can gallop. All tales about
infinityt and galloping numbers are completely irrelevant, and there is no
sense in which x really "becomes infinite." The assertion (3.341) means
that f (x) is near L whenever x is large. We examine an example. Every-
one who has an appreciation of the magnitudes of the numbers 1/2,
1/416, 1/7,528,432, and 1/1020 must believe that 1/x is near 0whenever x
is large, that is,(3.342) lim 1 = 0.
X,w XTo prove this, let e > 0. Let N = l/e. Then the inequality
x01 <e
is valid whenever I < ex and hence whenever x > 1/e and hence when-
ever x > N. Thus when a positive number e is given, we are able to find
a number N for which the e, N assertion is true. Therefore, (3.342) is a
true assertion. It is equally easy to attach a meaning to the assertion
that f(x) is near L whenever x is negative and has a large absolute value.
The abbreviated version of this assertion is(3.343) lim f(x) = L.
__-MWe say that the limit as x approaches minus infinity of f(x) is L.
There are some important modifications of these ideas that should now
be easily understood. In case f(x) = 1/(x - a) and also in some other
cases, we can cheerfully assert thatf(x) is large whenever x is near a and
x > a. This assertion is abbreviated to(3.35) lim f(x) = Co.
s-.a+It means that to each number M there corresponds a a > 0 such that(3.351) f(x)>M (a<x<a+6),
t For those who are really interested in infinity, a remark appears at the end of the
problems of this section.