**TI83F* wW fWLIMITSfWdWGRYefhW a q'kR y%$(g++-?/?1?3@69C;M=j@BhEJN SLimitsRick Bailey1 Limit DefinitionLetfbeafunctiondefinedonanopenintervalcontainingc(exceptpossiblyatc)andletLbearealnumber.The&statement- T  $d`@   $ -́ 0 $  $ $ P@$ @ ? H@pPA "'PD  Tformallymeansthatforeach   `  "thereexistsa  8@  X   suchthat'. J???p?p??2@? AR/@?0?p ??;?Jimplies %ADp(@ 0($0H` Limit ConceptLetfbeafunctiondefinedonanopenintervalcontainingc(exceptpossiblyatc)andletLbearealnumber.The&statement- T  $d`@    $-́ 0$ $ $ P@$ @ ? H@pPA "'PDTinformallymeansthattheoutputoffunctionfmaybemadearbitrarilyclosetoLbymakingxsufficientlyclosetoc.Some Basic Limits1 :????$?`? ? ?$?-̃ ??$ ?$ ??$@?$?????PA? "?'?PD??> o 0-- ,2 B B_Dpo-/Some Basic Limits2 4$` $`-́?$$?$ $   PA "'PD   @ Some Basic Limits3 <$`@ @ $d-́?$$?$ $  PA "'PD $   ɀ@    qLimit Properties1Letcandbberealnumbers. _$` p`    @"$X-̐d$$D$D$D $3 pPA "'PD'_/ D   @ @@ 2@ "@  "@@"@@ "@@ wF` @  A A@ Limit Properties2Letcbearealnumber. F$ `  "$-̐$$D$D$D $Ђ3PA "'PDF :???p?>>$>$D0 30b?!?p?> #: C$d`@   $-́ 0$ $ $ P$    H@pPA "'PDC C  @  ,}` s! "!  "! "  "! q @ `  $Limit Properties3Letcbearealnumber. Z???$ >`  A4 " A3$A!4-̐$b@`!4$DB@@!3$DD@@!4$D 8@!3$@A !3Ђ3x#0A4 A3 |PA "'PD=??Z  ?$d`@  $-́ 08$ 8$ 8$ P$   H@pPA "'PD? C  @  ,}` s! "!  "! "  "! q @ `  $Limit Properties4Letcbearealnumber. #Vd@   0   P$ 8`    H@$p-̐$$$$@ 8PA "v'PD(H! @ 0?@ @K0=` HHHH!  @@  ? $8? @@`>@Ā@@@p@@@@F` @  0P )?,0provided V$`   $ -̆$$$ $D@$ $@ ǂ3b!PA "'PDcLimit Properties5Letcbearealnumberandnbeapositiveinteger. U$ s``   "@$`-̐$$D$D$D $Ђ3pPA "'PD Hq  C@hAHAA%=`And A$DA$DA$DA$D!C @ @@@p  ? $86Polynomial LimitsLetcbearealnumberandpbeapolynomialfunction. H$`  $-̃$$$$$$D $38PA "'PD 8 $$$ D "8 2Rational LimitsLetcbearealnumberandrbearationalfunction. F$@`   $v-̃P$$$($H @PA "'PD <@  Pd@p#!B< providedthat 0$  @@  ! ! ! @ T'/isdefined}Radical LimitsLetnbeapositiveinteger.Thefollowinglimitisvalidforallcifnisodd,andisvalidforc>0ifniseven. Ak I$-`I { 2$8-́0$H k$H ?$PI$LÑI{OcPA "h'yPDh{ %? @@CL199=Composite LimitsOIffandgarefunctionssuchthat W$`@ @ $-̆$$$$D$ $ǂ3b!PA "'PD W)and0 F$2@`T H $v-̀$$$($H$ @PA !'PB@F 2???? ???"??$?D?D?DA?A???? ??'2,/then J$d!`  @ @$>A -́ ĀA$ A$ A$ p@A?$ @A?F`? @?H  ?p?PA? "?'?PD???J !(0 2???? ???"??$?D?D?DA?A???? ??Trig Limits1 H$` $v-́-$$Ĉ$$($$HPA "'PD /F+))J)*)((L/ Trig Limits2 M$`  $ð-́"@$$$A@$"B@Æ`PA "'PD 8L @@qG)H$H"H)HqG@  Trig Limit 3 J??$?`? ? ?$,?-́r?$"?$ "?$" ?$S"?3????PA? "'8PD8??  @@@ @ g, @ G H H ' @@ ,provided  @@@ @ g, @ G H H ' @@$ -isdefined Trig Limit 4 K$ `  $ǎ-́"$$$$"PA "'PD  &$""$Q&,provided &$""$Q,&isdefined Trig Limit 5 L $`  $`-́$H?$$?$$$$HÇ PA "'PD  'r8D>@ @Dr8 ',provided 'r8D>@ @Dr8('isdefined Trig Limit 6 L $`  $`-́H?$$?$H$H PA "'PD  $r8RD@@RDr8 $,provided $r8RD@@RDr8($isdefined Theorem 1.7Letcbearealnumberandletf(x)=g(x)forall '|< @0@PD8'inanopenintervalcontainingc.If&thelimitofg(x)asx-approachescexists,thenthelimitoff(x)alsoexistsand E$d`@   $-́ 0$ $ $ P$   H@pPA "'PD E'/ Aw3y$k`} { :$I-̆$w$$3$Dy$ $kǂ3}b{!acPA "h'yPDh{Squeeze Theorem Ifh(x)  4*???`?>`???)?%f(x) 8   ` Cg(x)forallxinanopenintervalcontainingc,except possiblyatcitself,andif'/ @$`  $-́b$b$$$݁PA "'PDHand @$` d $b-́b$$$$݀PA "'PD-Hthen @H  $ `H H| $| -̀H $H>$$$݀PA "'PD@PTheorem 1.9a M9,Kr$A"`1" "  I"$s3-̀$$$$; @@$PD@f $@'@PD@ ,_'}_86{~~}~~U,,Theorem 1.9b \88v"$H$ B@` B0  B( "$HH$p-̀$$$$v@(@HPD@ $@'@PD@\  ,$,BBB~B~B}B~B$~U,,zContinuityAfunctioniscontinuousatcifthefollowingthreeconditionsaremet.&-1.f(c)isdefined2. 6$d`@   $-́ 0$ $ $ P$   H@pPA "'PD>exists!)03. 6$d`@   $-́ 0$ $ $ P$   H@pPA "'PD >[ 6`@@@"@ @ @"@@@0*6Limit ExistenceLetfbeafunctionandletcandLberealnumbers.Thelimitoff(x)asxapproachescisLifandonlyif M00 @ `C b@ B@ ?B@@ B@@ ?B@ BA( 9&0@0@h0" 'P$M =00 @ `C b@ B@ B@@ B@@ B@ BA( 9&0@0@h0" 'P$Infinite LimitsLetfbeafunctionthatisdefinedateveryrealnumberinsomeopenintervalcontainingc(exceptpossiblyatcitself).&Thestatement- V$d`@   $-́ 0'$ "$ "$ P'$   H@pPA "'PDV informallymeansthattheoutputofthefunctionfmaybemadearbitrarilylargebymakingxsufficientlyclosetoc.qInfinite Limit P1pLetcandLberealnumbersandletfandgbefunctionssuchthat V$d`@  $-́ 0'$ "$ "$ P'$   H@pPA "'PD'V/and W$`@ @ $-̆$$$$D$ $ǂ3b!PA "'PDW. Then'/ F$ `  "$-̐$$D$D$D $Ђ3PA "'PDF :??>p<<<<$<$<D0 30b?!?p?> :   s`  `s  Infinite Linit P2apLetcandLberealnumbersandletfandgbefunctionssuchthat V$d`@  $-́ 0'$ "$ "$ P'$   H@pPA "'PD'V/and W$`@ @ $-̆$$$$D$ $ǂ3b!PA "'PDW. Then'/ Z???$ >`  A4 " A3$A!4-̐$b@`!4$DB@@!3$DD@@!4$D 8@!3$@A !3Ђ3x#0A4 A3 |PA "'PD=??Z  s` ` s ,forL>0Infinite Limit P2bpLetcandLberealnumbersandletfandgbefunctionssuchthat V$d`@  $-́ 0'$ "$ "$ P'$   H@pPA "'PD'V/and W$`@ @ $-̆$$$$D$ $ǂ3b!PA "'PDW. Then'/ Z>?$ `>  A " A$A!-̐$b@`!>$DB@@!=$DD@@!$D 8@!>$@A !Ђ3x#0A A<<|@Ā@@@p@@$@@`F`  @  $-̀$$$$ @PA "=`' PD!  @@  y$ky $yBxB{BBqBB0B$/.8