§ 3.4. The Theory of Limits
3.4.1. The limit of a sequence.
Definition. 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.
х1,х2,х3,х4,…,хn,….
Example.
Definition. A number а is called the limit of a sequence if, for any >0, there exists a number N() depending on , such
for
n>N,
Notation:
.
Example.
.
Definition. 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
.
Notation:
2
а– а а+
х(а–;а+)
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.
Example.
.
Assigning integer values to
n,
we obtain:
etc.; i.e. this variable has no limit.
Definition. 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 –.
Example. xn=n2+1 ; as n, хn tends to infinity.
3.4.2. The limit of a function. Suppose that y=f(x) is a function defined on a domain D containing a point а: аD.
Definition.
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:
.
(1)
This definition is interpreted geometrically as follows:
b+ y=f(x)
f(а)
b-
0
a–
a a+
If х belongs to a neighborhood of the point а, then the value of the function remains in the strip between b– and b+ .
Example. Find the limit of f(x)=5x–1 as x2, and determine .
,
,
i.e.,
.
To find , we must find x from the inequality for the function and substitute it in the inequality for the variable.
Definition. 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. 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.
Remark 1. The right and left limits may not coincide.
Example.
Find the limit of the function
as
x
0.
Substituting,
we obtain
We were taught at school that division by zero is not allowed. But at this author’s University, it is allowed. So, consider the fractions
,
,
…,
.
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
,
.
Remark 2. A function may not exist at a point а, but its limit may exists (and equal infinity).
Example.
;
an
indeterminacy, although the limit exists:
.
Definition.
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:
.