reals,Shas an upperbound,c. 3) Thereare then threepossibilitiesto explore:f(c) <d,f(c) =d, orf(c) >d.
Exploretheseand showwhyf(c) =d.
InfiniteLimits
LearningObjectives
A studentwill be able to:
- Find infinitelimitsof functions.
- Analyzepropertiesof infinitelimits.
- Identifyasymptotesof functions.
- Analyzeend behaviorof functions.
Introduction
In this lessonwe will discussinfinitelimits.In our discussionthe notionof infinityis discussedin two contexts.
First,we can discussinfinitelimitsin termsof the valuea functionas we increasexwithoutbound.In this
casewe speakof the limit off(x) as xapproaches and write. We couldsimilarlyrefer
to the limit off(x) asx approaches - and write.
The secondcontextin whichwe speakof infinitelimitsinvolvessituationswherethe functionvaluesincrease
withoutbound.For example,in the caseof a rationalfunctionsuchasf(x) = (x+ 1)/(x^2 +1), a functionwe
discussedin previouslessons:
Atx= 1, we havethe situationwherethe graphgrowswithoutboundin both a positiveand a negativedirec-
tion. We say that we havea verticalasymptoteatx= 1, and this is indicatedby the dottedline in the graph
above.
In this examplewe note that doesnot exist.But we couldcomputeboth one-sidedlimitsas follows.