2. The Theory of Limits
The limit of a sequence.
Definition
11. A
sequence is an infinite set of terms, each of which is assigned a
number. The terms of a sequence must obey a certain law.
Example.
Definition
12. A
number
is called the limit of a sequence if, for any >0,
there exists a number N()
depending
on
, such
for
.
Notation:
.
Example.
.
Definition
13. The
limit of a variable
х
is
a number а
such that for any >0,
there
exists an х
starting with which all х
satisfy the inequalities
.
Properties:
1. The limit of a constant number equals this number.
2. A variable can not have two different limits.
3. Some variables have no limit.
Definition
14. We
say
that
х
tends to infinity if, for any number М,
there exists an х
such that, starting it,
.
(a) M>0, x>M, x;
(b) M<0, x<–M, x –.
Definition
15. A
number b
is
called the limit of the function f(x)
as
х
а
if, for any given >0,
there exists a small positive
depending on
(()>0)
such that, for any х
satisfying the inequality
,
.
Notation:
.
Definition
16. The
left limit of a function f(x)
as
x
a
is the limit of f(x)
as
x
a,
and х<а.
Notation:
.
Definition
17. The
right limit of a function f(x)
as
x
a,
is the limit of f(x)
as
x
a,
and х>а.
Notation:
.
If the left limit equals the right limit and some number b, then b is the limit of the function as x a.
The
denominator of the last fraction is very small, and the fraction is
very large:
.
Decreasing the denominator and, finally, divide 1 by 0, we obtain
infinity, i.e.,
Using this
relation, we find
,
.
Definition
18. A
number b
is called the limit of f(x)
as х
if, for any >0,
there exists a (large) number N
depending on
such
that
for
any
.
Notation:
.
Infinitesimals and bounded functions.
Definition
19. A
function f(x)
is
said to be infinitesimals as
х
а
if, for any М,
there exists a
such
that
whenever
.
Notation:
.
Definitions
20.1.
A function f(x)
is said to be bounded on a domain D
if, for any х
from D,
.
If this condition is not satisfied, then the function is said to be unbounded, i.e., this function is an infinite quantity.
20.2.
A function f(x)
is said to be bounded as
if, for any х
from a neighborhood of а,
.
20.3. A function f(x) is said to be bounded as х if, for any х>N, .
Theorem I. If a function f(x) has a finite limit as then f(x) is bounded as .
Theorem
II.
If a function has a limit as
and
this limit is not equal to zero, then
is bounded as
.
3. Fundamental Theorems on Limits
Theorem III. The limit of the algebraic sum of finitely many functions equals the sum of the limits of these functions:
.
Theorem IV. The limit of the product of two functions equals the product of the limits of these functions:
.
Theorem V. The limit of the ratio of two functions equals the quotient of the limits of the numerator and the denominator:
.
Theorem
VI (the
sandwich theorem). If, in a neighborhood of a point а,
the inequalities
u (x)
z (x)
v (x)
hold
,
then limit of z(x)
exists and equals
.
Theorem
VII. If
a function f(x)
is monotonically increases (decreases) and is bounded as х
а
above
(below), then f(x)
has a finite limit
.
The First Remarkable Limit and Its Generalization
The
following limit exists and equals 1:
.
The Second Remarkable Limit
Consider
the limit
.