Consider. Find.
We see that and note that property#5 doeshold.Henceby directsubstitutionwe
have
Example5:
Consider Thenwe havethatf(g(x)) is undefinedand we get the indeterminate
form 1/0. Hence doesnot exist.
Limitsof TrigonometricFunctions
In evaluatinglimitsof trigonometricfunctionswe will look to rely moreon numericaland graphicaltechniques
due to the uniquebehaviorof thesefunctions.Let’s look at a coupleof examples.
Example6:
Find.
We can find this limit by observingthe graphof the sine functionand usingtheCALCVALUEfunctionof
our calculatorto showthat.
Whilewe couldhave foundthe limit by directsubstitution,in general,whendealingwith trigonometricfunctions,
we will rely less on formalpropertiesof limitsfor findinglimitsof trigonometricfunctionsand moreon our
graphingand numericaltechniques.
The followingtheoremprovidesus a way to evaluatelimitsof complextrigonometricexpressions.
SqueezeTheorem
Supposethatf(x) ≤g(x) ≤h(x) forxneara, and.
Then.
In otherwords,if we can find boundsfor a functionthat havethe samelimit,then the limit of the function
that they boundmusthavethe samelimit.
Example7:
Find.