Limit of a Sequence:
If a sequence {} tends to a limit , then we say that is the limit of a sequence {}, and write
.
In other words,
“A real number is the limit of a sequence {}, if for every , there exists a positive integer (or N) such that,
implies that .”
The number is called the limit of the sequence {}, and we write, as , or
, or simply, .
Precise Definition of a Limit of a Sequence at Infinity –
“If for every , there exists a positive integer (or N) such that
implies that ,
then the number is called the limit of the sequence , and we write,
, or simply, .”
(Note: When it is said that this sequence has a limit, it means that the limit is unique, that is, a finite and definite real number.)
Limit of a Sequence Examples:
Example 1:
Show that the sequence {} has the limit 0.
Proof:
Here, , and .
In this example, we have to show that the sequence {} has the limit 0, that is,
.
To prove this limit: ; we consider , and show that there exists a positive integer such that,
implies that , that is,
implies that .
To show this, we must find a positive integer such that implies that,
, or
, or
, or
, or
.
Hence, if we take , then for every , there is a positive integer such that implies that .
Therefore, , or the sequence {} has the limit 0.
Example 2:
If , where c is a real number, prove that .
Proof:
Here, , and .
To prove this limit: ; we consider , and show that there exists a positive integer such that,
implies that .
Now, given, , we must find a positive integer such that implies that,
, or
, or
, or
.
Hence, if we take , then for all , we have .
Therefore, , or the sequence has the limit .
Example 3:
Use the definition of the limit of a sequence to show that the sequence where,
,
has the limit 3.
Proof:
Here, , and .
In this example, we have to show that the sequence has the limit 3, that is,
.
To prove this limit, we consider , and show that there exists a positive integer such that,
implies that , that is,
implies that .
To show this, we must find a positive integer such that,
, …… (1)
for all .
This inequality (1) is equivalent to
. …… (2)
Now,
,
then inequality (2) will hold if,
.
Hence, if we take , then for every , there is a positive integer such that implies that .
Therefore, , or the sequence where has the limit 3.
Tags: define the limit of the sequences, how to find limit, finding limits of sequences
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