In makingthis simplification,we are indicatingthat the originalfunctioncan be viewedas a linearfunction
for x valuescloseto but not equalto 1, that is,
for. In termsof our limits,we can say.
Example2:
Find.
This is anothercase wheredirectsubstitutionto evaluatethe limit givesthe indeterminateform 0/0. Reducing
the fractionas beforegives:
.
Example3:
.
In orderto evaluatethe limit, we needto recallthat the differenceof squaresof real numberscan be factored
as
We then rewriteand simplifythe originalfunctionas follows:
Hence.
You will solvesimilarexamplesin the homeworkwheresomecleverapplicationsof factoringto reduce
fractionswill enableyou to evaluatethe limit.
Limitsof CompositeFunctions
Whilewe can use the Propertiesto find limitsof compositefunctions,compositefunctionswill presentsome
difficultiesthat we will fully discussin the next Lesson.We can illustratewith the followingexamples,one
wherethe limit existsand the otherwherethe limit doesnot exist.
Example4: