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  an = | 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 n-th 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  an ® 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 {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
 Solution:
 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:
Functions contents E