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Our definition of tangent to a curve is going to be based on the idea of limit. The word limit was used in mathematics long before the
definition we
will give was thought of. One finds statements like `` The limit of a
regular
polygon when the number of sides becomes infinite, is a circle." Early
definitions
of limit often involved the ideas of time or motion. Our definition will be
purely
mathematical.
10.5
Definition (Interior points and approachable points.)
Let
be a subset of
. A
point
is an
interior point of
if there is some positive
number
such that the interval
is a subset of
.
A point
is an
approachable point from if there is some
positive
number
such that either
or
. (Without loss of generality we could replace ``
"
in this definition by
for some
.)
Note that interior points of must belong to . Approachable points of
need not belong to . Any interior point of is approachable from
.
10.7
Definition (Limit of a function.)
Let
be a real valued function such that
. Let
and let
. We say
|
(10.8) |
if
- 1)
- is approachable from
, and
- 2)
- For every sequence in
Note that the value of (if it exists) has no influence on the meaning of
. Also the `` " in (10.8) is a dummy
variable, and can be replaced by any other symbol that has no assigned meaning.
10.9
Example.
For all
we have
Also
and
by lemma
9.34. Also
by theorem
9.37.
10.10
Example.
is not defined.
Let
. Then
is a sequence in
, and
and
. We know there is no number
such that
.
10.11
Example.
Let
be the spike function
Then
,
since if
is a generic sequence in
, then
is the constant sequence
.
10.12
Example.
The limit
does not exist. If
, then the domain of
consists of the single point
, and
is not approachable from
.
If we did not have condition
1) in our definition, we would have
which would not be a good thing. (If there are no sequences in
, then
is true, no matter what
statement about
is.)
In this course we will not care much about functions like
.
10.13
Example.
I will show that
|
(10.14) |
for all
.
Case 1: Suppose
. Let be a generic sequence in
such that . Then
Now, since
, we have
so by the squeezing rule
which is equivalent to
This proves (
10.14) when
.
Case 2: Suppose .
The domain of the square root function is , and is approachable
from this set.
Let be a sequence in
such that . To show that
, I'll use the definition of limit. Let
. Then
, so by the definition of
convergence,
there is an
such that for all
we have
.
Then
for all
we have
and hence
.
Many of our rules for limits of sequences have immediate corollaries as
rules
for limits of functions. For example, suppose are real valued functions with
and
. Suppose
and
. Let be a generic sequence in
such that . Then
is a
sequence in
and , so
Also is a sequence in
and so
By the sum and product rules for sequences, for any
and
and thus we've proved that
and
Moreover if
(so that ), and if
for all (so that for all
),
it follows from the quotient rule for sequences that
so that
Actually all of the results just claimed are not quite true as stated. For we
have
and
but
The correct theorem
is:
Proof: Most of the theorem follows from the remarks made above. We will assume
the remaining parts.
10.16
Theorem (Inequality rule for limits of functions.) Let and be
real functions with
and
.
Suppose that
- i
-
and
both
exist.
- ii
- is approachable from
.
- iii
- There is a positive number such that
Then
.
Proof: Let be a sequence in
such that .
Then is a sequence in
that converges
to , so by the definition of limit of a function,
Similiarly
Also
for all , so it follows from the inequality
rule for limits of sequences that
,
i.e.
.
10.17
Theorem (Squeezing rule for limits of functions.) Let , and be
real functions with
,
,
and
.
Suppose that
- i
-
and
both
exist and are equal.
- ii
- is approachable from
.
- iii
- There is a positive number such that
for all in
.
Then
.
Proof: The proof is almost identical to the proof of theorem 10.16.
Next: 10.3 Definition of the
Up: 10. Definition of the
Previous: 10.1 Velocity and Tangents
  Index
Ray Mayer
2007-09-07