and.
Moreformally, we definetheseas follows:
Definition:The right-handlimit of the functionf(x) atx=ais infinite,and we write , if for
everypositivenumberk, thereexistsan openinterval(a, a + δ) containedin the domainoff(x), suchthat
f(x) is in ( ) for everyxin (a, a +δ).
The definitionfor negativeinfinitelimitsis similar.
Supposewe look at the function and determinethe infinitelimits
and.
We observethat asxincreasesin the positivedirection,the functionvaluestend to get smaller. The same
is true if we decreasexin the negativedirection.Someof theseextremevaluesare indicatedin the following
table.
We observe that the values are getting closer to Hence and
.
Sinceour originalfunctionwas roughlyof the form , this enablesus to determinelimitsfor all
otherfunctionsof the form with Specifically, we are able to concludethat
. This showshow we can find infinitelimitsof functionsby examiningthe end behavior of the function
The followingexampleshowshow we can use this fact in evaluatinglimitsof rationalfunctions.
Example1:
Find.
Solution:
Notethat we havethe indeterminateform,so LimitProperty#5 doesnot hold.However, if we first divide
both numeratorand denominatorby the quantityx^6 , we will then havea functionof the form