Theseare examplesof limitsof basicconstantand linearfunctions, and
We note that eachof thesefunctionsare definedfor all real numbers.If we applyour techniquesfor finding
the limitswe see that
and observethat for eachthe limit equalsthe valueof the functionat the -valueof interest:
Hence. This will also be true for someof our otherbasicfunctions,in particularall
polynomial and radical functions, provided that the function is defined at x = a. For example,
and. The propertiesof functionsthat makethesefactstrue
will be discussedin Lesson1.7. For now, we wish to use this idea for evaluatinglimitsof basicfunctions.
However, in orderto evaluatelimitsof morecomplexfunctionwe will needsomepropertiesof limits,just as
we neededlaws for dealingwith complexproblemsinvolvingexponents.A simpleexampleillustratesthe
needwe havefor suchlaws.
Example1:
Evaluate. The problemhere is that whilewe knowthat the limit of eachindividualfunction
of the sum exists, and , our basiclimitsabovedo not tell us what happenswhen
we find the limit of a sum of functions.We will statea set of propertiesfor dealingwith suchsophisticated
functions.
Propertiesof Limits
Supposethat and both exist.Then
- wherecis a real number,
- wherenis a real number,