(MalcolmKe:所有数学教科书讲微积分的时候都是先定义极限然后再定义连续性,我觉得既然
实数轴天然具有连续性,应该直接利用连续性来定义极限,利用数轴连续性证明单调函数
的连续性,再利用单调函数的连续性来证明其它函数的连续性)
Funnels: A more intuitive definition of limits
Abstract: We describe a different definition of "limit,"
which may be easier to understand than the definition given in
most calculus books, though it is equivalent to that usual definition.
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Funnels:
A More Intuitive Definition of Limit
by Eric Schechter,
Vanderbilt University
Our textbook, by Stewart, gives this imprecise definition:
lim_{x→x0} f(x)
= L
is defined to mean that
we can make the values of f(x) arbitrarily close to L (as
close to L as we like) by taking x to be sufficiently
close to x_{0}, but not equal to
x_{0}.

This imprecise definition has several possible
interpretations, and so the student is
left to guess which interpretation is the right one.
The student is aided by the several examples that
accompany the definition. For proofs, the imprecise
definition is inadequate, and ultimately a ``precise
definition'' must be given; Stewart does so 20 pages later.
Evidently he waited that long because he wished
to avoid intimidating the student; the precise definition
is much more complicated. It states that
lim_{x→x0} f(x) = L
means
for each number ε > 0
there exists a corresponding
number δ > 0
with the property that, whenever x is a
number with 0 < xx_{0} < δ, then
f(x)L < ε. 
I will refer to this as the epsilondelta definition. It is
the classical definition of limit (for a realvalued
function of a real variable). Of course, different limit
assertions may require different choices of epsilons and
deltas: We could use one system of epsilons and deltas
for the assertion
lim_{x→3} (2x5) = 1,
and another system
of epsilons and deltas for the assertion
lim_{x→0} (sin x)/x = 1.
Many (perhaps most) calculus students have difficulty
understanding and learning the epsilondelta definition of a
limit. I can state
several reasons why the epsilondelta definition is
difficult to understand (although the student doesn't need
to be aware of these reasons): it has too many
variables; it has too many nested clauses; it does not
suggest anything about a rate of convergence; and it
cannot be illustrated easily with a picture.
I sometimes tell my students to memorize the
epsilondelta definition, word for word; understanding will come
later (if at all). I caution the students to be careful with
their memorizing; students who do not yet fully understand the
definition may inadvertently change the wording slightly, in some
fashion that sounds inconsequential to the untrained ear but
greatly changes the mathematical content.
In the paragraphs below,
we shall consider an alternate definition, which may be
easier for students to understand; we will call it the
funnel definition of limits. (This experimental
approach is modified from some asyet unpublished
material by Professor Bogdan Baishanski of Ohio State
University.) The epsilondelta definition and the
funnel definition are equivalent, as we
shall demonstrate at the end of this document. The
proof of equivalence requires some understanding of
subsequences, but no other specific knowledge beyond
calculus.
The funnel definition of ``limit'' is in two steps; first we must
define a ``funnel.''
Definition. A funnel is
an increasing function φ,
whose
domain and range are of the form (0,a) and (0,b)
respectively, where a and b are any members of
(0,+∞].

Examples.
φ(t) = t,  considered
as a function from (0,+∞)
to (0,+∞)

φ(t) = t,  considered
as a function from (0,2) to (0,2)

φ(t) = t^{2},  considered
as a function from (0,+∞)
to (0,+∞)

φ(t) = √t,  considered
as a function from
(0,+∞)
to (0,+∞)

φ(t) = t/(1+t),  considered
as a function from
(0,+∞)
to (0,1)

φ(t) = tan(t),  considered
as a function from
(0,π/2)
to (0,+∞)

φ(t) = 1cos(t),  considered
as a function from
(0,π/2)
to (0,1)

I have chosen the name "funnel" to emphasize that the
graph of
φ(t)
is narrow (i.e.,
φ(t)
is small) for t near 0, like a kitchen
funnel turned on its side; this is perhaps best illustrated
by the funnel
φ(t) = t^{2}.
That property is not
immediately evident in the graph of
√t,
which appears to
be flat at its left end, rather than pointed.
Nevertheless, √t
is an increasing function, so
√t
is indeed a funnel.
Actually, the functions t, t^{2},
√t
will suffice for most
early applications. The beginning student does not need
to be burdened with exercises such as "prove that
ln(1+t)
is a funnel." However, such proofs are not
particularly difficult: 1+t is an increasing function, and
ln is an increasing function, hence ln(1+t) is an
increasing function. Moreover, it may be useful to
mention at the outset that a wide variety of functions
can be used as funnels, and that particular funnels can
be devised for particular applications.
Observation. Suppose that φ
is a funnel, with
domain (0,a) and range (0,b). Suppose that
0 < a_{1} < a, and
let b_{1} = φ(a_{1}).
Then φ,
considered as a
function from (0,a_{1}) to (0,b_{1}), is also a funnel.
Summarizing this result a bit imprecisely, we say that
any funnel, restricted to a smaller domain, is also a
funnel. That fact is important for some applications.
Moreover, in most applications, it doesn't matter what
size domain (0,a) we use, as long as a > 0
and a is small enough to satisfy the particular
application; generally any smaller positive number will also
work. Consequently,
in many applications, we don't bother to specify the
domain.
Definition. The equation
lim_{x→x0} f(x) = L
(in the sense of funnels) means that
there exists a funnel φ
with the property that,
whenever x is a real number such that
xx_{0} is in the domain of φ,
then x is in the domain of
f and
f(x)L <
φ(xx_{0}).

Of course, different limit assertions may require
different funnels. For instance:

lim_{x→3} (2x5) = 1.
Proof:
Take φ = 2t.

lim_{x→0} (sin x)/x = 1.
Proof:
Take
f(x) = (sin x)/x for x≠0,
x_{0}=0, L=1, and
φ = 1cos(t)
for
0 < t < π/2.
Using geometry, we can prove that
cos(x) < x^{1}sin(x) < 1
when 0 < x < π/2
(see Stewart, page 171; no one claimed that the proof
would be obvious!). Hence
0 < 1 
x^{1}sin(x) < 1  cos(x)
when 0 < x < π/2
Note that cos(x)
is an even function of x. It follows that
f(x)L = 1 x^{1}(sin x)}
< φ(xx_{0})
when xx_{0} is in the
domain of φ.