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 