numbersas we get closeto Alternatively, for we see that the pointsall lie in the fourth
quadrantand decreaseto largenegativenumbers.If we inspectactualvaluesvery closeto we can
see that the valuesof the functiondo not approacha particularvalue.
ERROR
For this example,we say that doesnot exist.
FormalDefinitionof a Limit
We concludethis lessonwith a formaldefinitionof a limit.
Definition.We say that the limit of a function at is writtenas , if for everyopen
interval of thereexistsan openinterval of that doesnot include suchthat is in
for every in
This definitionis somewhatintuitiveto us giventhe exampleswe havecovered.Geometrically, the definition
meansthat for any linesy=b 1 ,y=b 2 belowand abovethe liney=L, thereexist verticallinesx=a 1 ,x=
a 2 to the left and right ofx=aso that the graphoff(x) betweenx=a 1 andx=a 2 lies betweenthe linesy
=b 1 andy=b 2. The key phrasein the abovestatementis “for everyopenintervalD”, whichmeansthat
evenif D is very, very small(that is,f(x) is very, very closeto L), it still is possibleto find intervalN where
f(x) is definedfor all valuesexceptpossiblyx=a.
Example2:
Use the definitionof a limit to provethat