3.
4.
- providedthat
With thesepropertieswe can evaluatea widerangeof polynomialand radicalfunctions.Recallingour ex-
ampleabove,we see that
Find the followinglimit if it exists:
Sincethe limit of eachfunctionwithinthe parenthesesexists,we can applyour propertiesand find
Observethat the secondlimit, , is an applicationof Law #2 with. So we have
In mostcasesof sophisticatedfunctions,we simplifythe task by applyingthe Propertiesas indicated.We
wantto examinea few exceptionsto theserulesthat will requireadditionalanalysis.
Strategiesfor EvaluatingLimitsof RationalFunctions
Let’s recallour example
We saw that the functiondid not haveto be definedat a particularvaluefor the limit to exist.In this example,
the functionwas not definedforx= 1. Howeverwe wereable to evaluatethe limit numericallyby checking
functionalvaluesaroundx= 1 and found.
Notethat if we tried to evaluateby directsubstitution,we wouldget the quantity0/0, whichwe referto as
anindeterminateform.In particular, Property#5 for findinglimitsdoesnot applysince
. Hencein orderto evaluatethe limit withoutusingnumericalor graphicaltechniqueswe makethe following
observation.The numeratorof the functioncan be factored,with one factorcommonto the denominator,
and the fractionsimplifiedas follows: