Calculus: Analytic Geometry and Calculus, with Vectors

(^126) Functions, limits, derivatives
that e is small, we need not deny ourselves the satisfaction of the feeling
that, when the given a is small, the dotted lines must be close together
and the S must be small.
For the case in which f(x) = x2 and a = 3, we have been discussing
questions involving values of f(x) when x is near a. Our serious interest
often lies in such questions when f(x) has a more complicated expression,
say one of
1 2+x- VJ_ sinx
(1 -x)110.
X X x
We should therefore know that the assertions in the four boxes
(3.25)
(3.251)
(3.26)
(3.27)
f(x) is near L whenever x is near a but x 0 a.
f(x) is a good approximation to L whenever
x is a good approximation to a but x 5A a.
To each e > 0 there corresponds a h > 0 such that
lf(x) - LI < e whenever 0 < Ix - at < S.
lim f(x) = L.
x-.a
have identical meanings. When we have plenty of time, we can always
replace the epsilon-delta assertion by the following more ponderous but
psychologically satisfying one. To each positive number a there corre-
sponds a positive number S such that f (x) approximates L so closely that
If(x) - LI < e whenever x is different from a but approximates a so
closely that Ix - at < S. The assertion (3.27) is read "the limit as x
approaches a of f(x) is L."
If f and a are such that there is no L for which the four assertionsare
true, then we say that
lim f(x)
does not exist. Complete comprehension of this matter is essential;
otherwise, we must be eternally confused bya statement that a thing at
which we are looking does not exist.
Some assertions involving limits are not completely simple. There
will come a day when we must know there isa number e, having the
approximate value in
(3.271) e = 2.71828 18284 59045,