TheoremIf a sequenceis convergent,then its limit is unique.Keepin mindthat beingdivergentis not the sameas not havinga limit.
L’Hôpital’s Rule
Realistically, we cannotgrapheverysequenceto determineif it has a finitelimit and the valueof that limit.
Nor can we makean algebraicargumentfor the limit for everypossiblesequence.Just as thereare indeter-
minateformswhenfinding limitsof functions,thereare indeterminateformsof sequences,suchas
. To find the limit of suchsequences,we can applyL’Hôpital’s rule.
Example 10
Find.
Solution
We solvedthis limit by usinga graphin Example5. Let’s solvethis problemusingL’Hôpital’s rule. The nu-
meratoris ln(n) and the denominatorisn. Bothfunctionsy= ln(n) andy=ndo not havelimits.So, the se-
quence is of the indeterminateform. Sincethe functionsy= ln(n) andy=nare not differen-
tiable,we applyL’Hôpital’s rule to the correspondingproblem, , first. Takingthe first derivative
of the numerator and denominator of , we find. Thus,
becausethe pointsof are a subsetof the pointsof the function
asxapproachesinfinity. We also confirmedthe limit of the sequencewith its graphin Example5.
Rules,Sandwich/Squeeze
Propertiesof functionlimitsare also usedwith limitsof sequences.
Theorem(Rules)Let {an} and {bn} be sequencessuchthat and. Letcbe any constant.Thenthe followingstatementsare true: 1. The limit of a constant
is the sameconstant.2. The limit of a constanttimesa se-
quence is the same as the constant times the limit of the sequence. 3.
The limit of a sum of sequencesis the sameas
the sum of the limitsof the sequences.4. The
limit of the productof sequencesis the sameas the productof the limitsof the sequences.5. IfL 2 ≠
0, then. The limit of the quotientof two sequencesis the same
as the quotientof the limitsof the sequences.