Limits of Sequences, Lim
We already know what are arithmetic and geometric progression 
a sequences of values. Let us take the sequence a_{n} = 1/n, if k
and m are
natural numbers then for every k < m is true a_{k} > a_{m}, so as
big as it gets n as smaller is becoming a_{n}
and it's always positive, but it never reaches null. In this case we say that 0 is
the lim a_{n>∞}
if n>∞, or the other
way to write it down lim_{n>∞} a_{n} = 0.
Limit Definition
The number a is called limit of a sequence, if for every ε > 0 it can be found a number n_{ε}, so that for all the members of the sequence a_{n} with index n > n_{ε} is true that a  ε < a_{n} < a + ε.
Basic rule
A sequence not always has limit, and sometimes has unreal limit( ∞ or +∞ ). The limits +∞ and ∞ are called unreal limits.
If the sequences a_{n} and b_{n} both have real limits then,
the sequences
a_{n} + b_{n},
a_{n}  b_{n}, a_{n}.b_{n} and a_{n} / b_{n} also have real limit and:
lim_{n > ∞}(a_{n}  b_{n}) = lim_{n > ∞}a_{n}  lim_{n > ∞}b_{n}
lim_{n > ∞}(a_{n} . b_{n}) = lim_{n > ∞}a_{n} . lim_{n > ∞}b_{n}
lim_{n > ∞}(a_{n}/ b_{n}) = lim_{n > ∞}a_{n} / lim_{n > ∞}b_{n}
if b_{n} ≠ 0 and lim_{n>∞}b_{n} ≠ 0
If a_{n} < b_{n} for every natural n
and lim_{n>∞}a_{n} = a,
lim_{n>∞}b_{n} = b
then a ≤ b
If a_{n} ≤ b_{n} ≤ c_{n} or every real
n and if lim_{n>∞}a_{n} = lim_{n>∞}c_{n} = A
then lim_{n>∞}b_{n} = A.
If a_{n} ≥ 0 and lim_{n>∞}a_{n} = a, then the sequence b_{n} = √a_{n} also has a limit and lim_{n>∞}√a_{n} = √a_{n}.
If a_{n} = ^{1}/_{nk} and k ≥ 1 then lim_{n>∞}a_{n} = 0.
(1+1/n)^{n} < e < (1 + 1/n)^{n1}
e is the number of Neper.
If sequence a_{n} has a unreal limit( ∞ or +∞ ) then the sequence 1/a_{n} has a limit and lim_{n>∞}^{1}/_{an} = 0
If sequences a_{n} and b_{n} have unreal limits and lim_{n>∞}a_{n}=+∞, lim_{n>∞}b_{n}=+∞ then:
lim_{n>∞}(a_{n} . b_{n}) = +∞
lim_{n>∞}a_{n}^{k} = +∞ if k > 0
lim_{n>∞}a_{n}^{k} = 0; if k < 0
lim_{n>∞}a_{n} = ∞
Lim problems
Exercise 1:
If a^{n} = 5.4^{n}, lim_{n>0}a_{n} = ?
Answer:
lim_{n>0}a_{n} = lim_{n>0}5 . lim_{n>0}4^{n} = 5 . 4^{0} = 5.1 = 5
Exercise 2:
If a_{n} = 

then lim_{n>∞an = ?} 
Answer:
lim_{n>∞} 

= lim_{n>∞} 

. 

= lim_{n>∞} 

= 3 
Exercise 3:
If lim_{an>1} = 

= ? 
Answer:
lim_{an>1} = 

=  lim_{an>∞} 

= 
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