|
|
Sequences
and limits
|
|
Operations
with limits
|
We
usually use following results when finding the limit of a sequence. |
Let
{an}
and {bn}
be two sequences of real numbers such that an
®
a
and bn
®
b.
Then, |
|
|
|
|
|
|
|
|
Operations
with limits examples
|
Let
apply the above operations with limits to calculate limits of given
sequences.
|
Example:
Find |
|
|
|
|
Example:
Find |
|
|
Solution: |
|
|
First factor the term of highest degree from both the
numerator and denominator. |
Note
that the
same procedure can be applied to every fraction for which the numerator
and denominator are polynomials in n.
|
The limit of such a
fraction is the same as the limit of the quotient of the terms of
highest degree. |
|
Example:
Find |
|
|
Solution: |
|
|
|
|
The
limit of a function
|
The
definition of the limit of a function
|
The
limit of a function is a real number L
that f
(x)
approaches as x
approaches a given real number a, |
|
written |
|
|
if
for any e
> 0 there is a d
(e)
> 0 such that |
f
(x)
-
L | < e
whenever | x
-
a | < d
(e). |
The
definition says, no matter how small a positive number e
we take, we can find a positive number d
such that, for an arbitrary chosen value of x
from |
the interval a
-
d
< x
< a +
d, |
the corresponding function's values lie inside |
the interval
L
-
e
< f
(x)
< L +
e, |
as shows the right figure. |
That
is, the function's values can be made arbitrarily
close to the number L
by choosing x
sufficiently
close to
a, but not equal to
a. |
|
|
|
Therefore,
the number d,
that measures the distance between a point x
from the point a
on the x-axis,
depends
on the number e
that measures the distance between f
(x)
from the point L
on the y-axis. |
|
Example: Given |
|
|
whenever |
|
|
A
limit is used to examine the behavior of a function near a point
but not at the point. The function need not even be defined at
the point. |
|
A
limit on the left (a left-hand limit) and a limit
on the right (a right-hand limit)
|
The
limit of a function where the variable x
approaches the point a
from the left or, where x
is restricted to values
less than a,
is written |
|
The
limit of a function where the variable x
approaches the point a
from the right or, where x
is restricted to values
grater than a,
is written |
|
If
a function has both a left-handed limit and a right-handed limit
and they are equal, then it has a limit at the point. Thus,
if |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Contents
H
|
|
|
|
|
|
Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved.
|
|
|