

The limit of a sequence,
theorems 
Some
important limits 
Operations
with limits 
Operations
with limits examples 






Some
important limits 
(1)
Let examine convergence of the sequence given by a_{n}
=  a ^{n} 
a) if  a 
> 1 then we can write 
a  = 1 + h,
where h
is a positive number. 
So,
by the binomial theorem 

If
we drop all terms beginning from the third that are all positive since
the binomial coefficients are natural numbers, if n > 2
and h
> 0, the right side become smaller, so obtained is the Bernoulli's
inequality 
(1 + h)^{n}
> 1 + nh,
n > 2. 
When
n
®
oo
then 1 + nh
tends to the positive infinity too, since we can make 1 +
nh
greater than any 
given positive number N,
if only we take 

therefore will even more tend to infinity
 a ^{n}
which is 

greater.
Thus, 

or 




b) if
0 <  a  < 1 then we can write 



Since b
> 1 then 

such that  a ^{n}
< e
whenever 

however small e is, 

this
inequality can be satisfied by choosing n
large enough. Therefore, 

or 




(2) Let examine convergence of the sequence given by 


The
sequence 

the nth
term of which we can write as 


For
every  a  >
1 there exists a natural number m
such that m <
 a  < m + 1
and n
> m
then 

since 

it follows that a_{n}
® 0
or we write 



(3) Let examine convergence of the sequence given by 


a) If a
> 1 then the sequence 

is decreasing, that is 


Let 

then by the Bernoulli's
inequality a
> 1 + nh
so that 


Since the numerator a

1 is fixed number then,
if n ® 0
then h ® 0
too, therefore 


So we
can write 



b) If
0 < a < 1 then the sequence 

is increasing, that is 


For
example, 



If
we write 



so it
follows that 

Therefore,




c) If
a = 1 then 




Since
in all three cases above, a), b) and c) we've got the same result,
then we can write 


Operations
with limits 
We
usually use following results when finding the limit of a sequence. 
Let
{a_{n}}
and {b_{n}}
be two sequences of real numbers such that a_{n}
®
a
and b_{n}
®
b.
Then, 








Operations
with limits examples

Let
apply the above operations with limits to calculate limits of given
sequences.

Example:
Find 





Example:
Find 





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 














Calculus
contents B 



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