CK-12-Calculus

8.1. Sequences http://www.ck12.org

Rules, Sandwich/Squeeze

Properties of function limits are also used with limits of sequences.
Theorem (Rules)
Let{an}and{bn}be sequences such that limn→+∞an=L 1 and limn→+∞bn=L 2.
Letcbe any constant. Then the following statements are true:

1. limn→+∞c=c
   The limit of a constant is the same constant.


2. limn→+∞c×an=c×limn→+∞an=cL 1
   The limit of a constant times a sequence is the same as the constant times the limit of the sequence.


3. limn→+∞(an+bn) =limn→+∞an+limn→+∞=L 1 +L 2
   The limit of a sum of sequences is the same as the sum of the limits of the sequences.


4. limn→+∞(an×bn) =limn→+∞an×limn→+∞bn=L 1 L 2
   The limit of the product of sequences is the same as the product of the limits of the sequences.


5. IfL 26 = 0 ,then limn→+∞

(an

bn

)

=limlimnn→→++∞∞bann=LL^12.

The limit of the quotient of two sequences is the same as the quotient of the limits of the sequences.
Let’s apply these rules to help us find limits.
Example 11
Find limn→+∞ 9 n^7 +n 5.
Solution
We could use L’Hôpital’s rule or we could use some of the rules in the preceding theorem. Let’s use the rules in the
theorem. Divide both the numerator and denominator by the highest power ofnin the expression and using rules
from the theorem, we find the limit:

n→lim+∞ 9 n^7 +n 5 =n→lim+∞

(^7) nn
(^9) nn+ (^5) nDividing both numerator and denominator byn
=n→lim+∞( 9 +^75
n
)Simplifying
=limlimn→+∞^7
n→+∞

( 9 + 5

n

)Applying the division rule for limits.

=lim limn→+∞^7

n→+∞^9 +limn→+∞^5 n

Applying the rule for the limit of a sum to the denominator

= 9 +^70 =^79 Evaluating the limits

Example 12
Find limn→+∞(^11 n−n^82 ).
Solution