Themulandeightheerenm CorSqueezeTheorem

Hereis areallytrickylimit 41 X'sin f Howdowecalculateit Thefunction y xcos Iisnotcontinuous at x O sowecan'tjustplugin X 0

Instead recallthat I Esin101E l forall ocIR TakeO TI ESinH e l

Nowmultiplythroughbyx2 x2 e x2sinIx EXZACx B

Notethatthe E doesnotchangedirectionwhenmultiplyingbyx2 becausex2 oThisshowsthatourfunction isinHx is sandwiched betweenthetwofunctionsAX XZandBCN XZ whoselimitsare

diffACx o noBan

Butsince itsinks is sandwiched inbetweenthesetwolimits theonlypossibility isainfox2sin xt O

Thisstrategyof sandwiching a function tofind a limitis calledthesandwichtheoremBCH

f

A11

SANDWICHTHEOREM LetAxlbeafunctionandlet acIR SupposethatAxlBallaretwofunctionssuchthat

AxlE fCHEBHI forall neara andfifaHx maBAI L

Then finnafail LThistheoremismostusefulwhendealingwithsineandcosinelimits

CosExemple Let'sevaluate 7 Startingfrom leSinane1 andmultiplyingthroughbyHx notingthatxcosincex isgoingto x wegettheinequalities

if 3 0 13 a theinequalitygotreversedbecause KoSincethetopandbottomofthesandwich namely and I gotoZeroas x no itfollowsfromtheSandwichTheoremthat

III 05 o

PROIP ifyouseea limitinvolving atrigfunction it is verylikelythattheSandwichTheoremwillhelpyou

Example let'scalculatethelimit aInfo Xesin Onceagainstartfrom I e sinH1etThenapplyinge to allsidesresultsintheinequality

e eesin e et aNTEetxpoientialspreserveMJ

Nowthingsgetsubtle You'reprobablytemptedto inequalities iemultiply thisinequalitythroughby likethis ey b e b

e x e esin e ex ifbsButthisonlyworkswhenxzo Inthiscase allthe SandwichTheoremgivesus istherightsidelimit lim

Koo Xesink

0

If co it's a differentstory multiplyingH1 resultsintheinequalitye x yesinit 3 ex

which onceagainbytheSandwichTheorem resultsinftp.xesinl o Great Sothelimitsfromeachsideexistandareequal

oXesinks

Iim