**TI89**MAINAppVariable file 07/28/04, 21:12RlimitsßY¥ZY…óG¿§Qcv†Š5S Ÿ ö \JÔ¤!!)Ñ+/é0357v:_?ò@DGuIúN:S®WLimProps&TheoremsSean Bird for TI89updated 3/17/05version 2Limit Definition Let f be a function  defined on an open interval containing c (except possibly at c) and &let L be a real number.The /^statement 0T  $d€€`¨@   $³ ì -Ì 0à $ˆ‚  $ˆ‚ à$ˆ‚ P@$ˆ„ @ÝĘ ?À H@p€PA€ "€'äPD€  8^formally A^meansPí W € €`€€ €ð€€€à that for each b 8€@ € €X €ˆ€˜€ð  there exists a #?J???p?ˆp?ˆ€?ˆ 2€@?ˆ AøR°/ˆ@‚?ˆ ’0?p âà??;?&such that 72A‚à‹Dp’„º(€„€@’‡à„ 0’€„À¢(€$0¢H€ä€`¡À‚€/8}impliAesPPLimit Concept Let f be a function  defined on an open interval containing c (except possibly at c) and %let L be a real number.The .^statement 0T  $d€€`¨@    $³ ì-Ì 0à$ˆ‚ $ˆ‚ à$ˆ‚ P@$ˆ„ @ÝĘ ?À H@p€PA€ "€'äPD€7^informally @^meansPthat the output of  function f may be made arbitrarily close to L by making x sufficiently %close to c.PâSome Basic Limits1 :????$?`? ? ?$³À?-̃ ?À?$ˆ„ ?$ˆ„ ?À?$ˆ„@?$ˆˆÀ?ÝÇ????PA€? "€?'ä?PD€??>P o 0-- ,2 B B_DŒpo-/PÇLim of identity 4$` $³`-Ì€?À$ˆ$ˆ?À$ˆ‚€ $ˆ„€ ÝÌÀ  PA€ "€'äPD€P   À@ €PâSome Basic Limits3 <$À`@ @ €$³dÀ-Ì€?À$ˆ$ˆ?À$ˆ‚€ $ˆ„€ÝÌÀ  PA€ "€'äPD€P $  € €€ É€@   € PtLimit Const*funLet c and b be real  numbers.i _$` €p`    @"$³Xô‚-Ìd$€$ˆ„D$ˆ„D€$ˆˆD $ˆ‘„ÝÐà‚3‚ €pPA€ "€'äPD€i%iP D  €  €@€ €@€À’̃°@ ·2€À@ ’"€€@  ’"€€@@’"@@ À’"‚@@ ÿwF`€ @€  ÀA€ŠŸ A@ P¼Limit of sumLet c be a real number. v lâ€@0$€ @@ 0; ôˆÈ $ˆ $?€ˆ!üDˆ „ ?€ˆ$ ;Üf@ ‚3@bP€! `À€@  ÿ@P`P  wá€@€0"@@     0;E˜È;È Îdˆ ˆDDˆˆþDDˆDDˆ$DD$;Üf îîf  !„ P@ ‚@`€€ ‚  ÿ@  P( `0PDLim of productLet c be a real number. ‚  y@‚€€ 8€„@  DÆxÆ„1„Üc(! øÌc 000A Ìc 00B Ìc 0< øÌc 00 Ìc@J@„=þ÷€@Œxc@ †€ €@øƒD?èuJ ŒP qဠ@ 0"@€   0v!fòÄuÈc™"ˆ!"ˆ‚!Bˆ(!‚ˆH! ;ÜÌ s»‡ˆ aP@ €`€À@  ƒ ÀDuÿ@Ÿèu P@Ju `@Œ {{P¼Lim of quotient #Vd€¨@   ì 0   P$ 8`˜    H@$³p€-Ìð$ˆ$ˆÿÿÿð$ˆ$ˆÝÐ@ 8PA€ "€Ðv'äPD€(HÌ!ˆ „ @Let c be a real number.PÇ 0?@ @€€K0=`€ÜÈ €€Hˆ€Hˆ€Hˆ€€Hˆ!€€ýÜ ŒÁ €@@€ ‚ ? ‚$€8?ÿÿÿÿÿø€ @€@€–`>ƒ°@¹Ä€À@‘„€€@‘ˆ€€@‘p@@‘€‚@@û¸ðF`€ @€ ø 0Pü€ à U,% 38V$€À`   $³ô‚ -̆$ü$ˆ„$ $ˆ„D@$ˆƒ„ ü$ˆ„@ ÝÇ‚3€Àb!À€PA€ "€'äPD€.7˜p@rovidedPPfLim of powerLet c be a real number and  n be a positive integer._ U$ €s``    "@$³ô‚`-Ì$à$ˆD$ˆDà$ˆD $ˆ„ÝЂ3‚€pPA€ "€'äPD€P Hq  ØC@hA€HA€A%˜=`„˜And €„A$D„A$D„A$D€„A$D!€„Cþî ŒÁ@ €@@@p€ ‚ ? ‚$€8PCPolynomial LimitsLet c be a real number and  p be a polynomial Qfunction. H$€`  $³Ä‚-̃$ü$ˆ†$$ˆ„$ü$ˆ„D $ˆˆ„ÝÏ38€PA€ "€'äPD€QP 8€ Ä$$$ D „"8€ P?Rational LimitsLet c be a real number and  r be a rational function.P F$@`   $³Ðv-̃Pð$ˆ†$ˆ„ð$ˆ„($ˆˆHÝÈÌ @PA€ "€'äPD€P <€@  ÐPd@€€ˆp#!B€ FF #$ € @@ ø  € ! ! ! à@€provided that %-.-7is definedPŠRadical LimitsLet n be a positive  integer. The following limit is valid for all c if n is odd, and is valid %for c > 0 if n is even..K -Ak þI$-`I { 2$³ì8-ÌÈ0þ$ˆH k$ˆH þ?$ˆ€PI$ˆƒLÝјI{ÐOàcPA€à "€h'äyPD€h{P %?À @@C€L€ˆ199PKComposite Limits¥ %‚8 H@@€ðf}bp@ ÿ€‰@@@‰€@ÿ qÿƒ€@@€@( ÿ€ àL ð„ÂF€@ð 0–8°EÈ|ˆ@ˆ@ˆQDˆ»¹Ü5 ŒŒŒ%.7@PInformal words: IF f(g(x))  is close to L when g(x) is close to k, and if g(x) is close to k when x is close %to c, thenPÚ Cl @ 8@‚€!1€x a çÀ‚ø cÀ  ††cÀ !†cÀ †øcÀ †cÀ@ ¡<÷½à@<Á@À€ƒ€|€A€"€?ô$€ G€à@  €  @‚€‚€A&0ð†0A8ã @Ac!€a€cA€€a€c€€a€c€a€c‚PA¼÷¼Œ`A0À@‚  @@‚€€Á  À @AÀ €ò °" ÀÂ/üÀ PÂ$  !@ À€€ vvv&v/v8vAvP(Informal words continued)  then... f(g(x)) is close to L when x is close to c.PôTrig Limits1 H$` $³ä°v-Ì-Èü$ˆˆ$ˆ€Äˆü$ˆ€$ˆ($ˆ$ˆHÝÁÏÜÌPA€ "€'äPD€P /FÊÈâ+)Š)J)*)Ê(â UUP Trig Limits2 M$`  $³àÁð-Ì"@Àà$ˆ€$ˆ€€à$ˆA@$ˆ"B@ÝÀàÆ`PA€ "€'äPD€P 8L @@qÆGŠ)H‚$H‚"HŠ)HqÆG@ P Trig Limit 3 J??$?`? ? ?$³Î,€?-Ìrÿ?$ˆ"?$ˆ "ÿ?$ˆ" ?$ˆS"?ÝÁ÷3????PA€? "€'ä8PD€8??P  @@@€ @€ g,Ž @²‘ G¢ H¢ H¢‘ '¢Ž € @@ ),  @@@€ @€ g,Ž @²‘ G¢ H¢ H¢‘ '¢Ž € @@provided)%).is definedP Trig Limit 4 K$ `  $³àÇŽÀ-Ì"€$ˆ$ˆ€$ˆ$ˆ"‰ÝÀàÀPA€ "€'äPD€ P &ÑÄ¢’$"’"’¢’$QÄ 0,0 #&ÑÄ¢’$"’"’¢’$QÄprovided%0.0is definedP Trig Limit 5 L $`  $³ã‡`-Ì$H€?À$ˆÈ$ˆ€Ä?À$ˆ€$€$ˆ$H„€ÝÁÇ ÀPA€ "€'äPD€ P 'r8¢ŠD>‚@ ‚@¢ŠDr8 1, 'r8¢ŠD>‚@ ‚@¢ŠDr81provided1is definedP Trig Limit 6 L $`  $³áÇ`-ÌH€?À$ˆ$ˆˆ?À$ˆH€$ˆH„€ÝÀ㇠ÀPA€ "€'äPD€ P $Œr8RŠD‚@‚@RŠDŒr8 -, $Œr8RŠD‚@‚@RŠDŒr8-provided-is definedPTheorem 1.7Let c be a real number and  let f(x) = g(x) for all 1 '€|à<€ @0à@PDØ81in an open &interval containing c. /If the limit of g(x) as x 8approaches c exists, thenPthe limit of f(x) also  exists and‘ ˆ@ €€@ @@ @E˜x v ³>@v Îd  1Ì€Ä@ DD  ˆ€„@ DD  ˆ€ˆ@ DD ( ˆ€p@( DD@ H ˆ€€@H îî@Ì@¹ÝÀð Ì@@@ @€€€øÐ ` `€ Gùÿ  @( !€0&‘/‘P Squeeze Theorem-  @"€$   @€€ `@€@ @xd q€à@ øC@ „€‚A  @@   $€€À @À  $€`@  @@  €€@ À@@  $€À @À  B¡(@!(@xbÁü†0@À&0F@€ p‚€ @ 0Àfor all x in an open "interval containing c, +except possibly at c 4itself, and if=P @$`  ô’$³’-Ìb$ˆb$ˆ’$ˆ€ô’€$ˆ€ÝâPA€ "€'äPD€J JJ $@$` d’ ”’$³b-Ìb$ˆ’$ˆ’€$ˆ€$ˆ€Ý€PA€ "€'äPD€and%J.JthenP @H  $ `ÒH ’H| ‘ˆ $³‘ˆ| -Ì€’H $ˆ€’H>$ˆ€$ˆ€$ˆ€Ý€PA€ "€'äPD€ JJPRTheorem 1.9a M9,€Kr$A"`1" "  I"$³s÷3-Ì€à$ˆ€$ˆÿÿÿðà$ˆ€$ˆ€ÝÀ; €@@$PD@f $@'ä@PD@€P ,_'}_86{~~}~~U,,PContinuityA function is continuous  at c if the following three conditions are met.&/8P¤ 6$d€`¨@   $³ ì-Ì 0$ˆ‚ $ˆ‚ $ˆ‚ P$ˆ„ ÝĘ  H@p€PA€ "€'äPD€1. f(c) is defined 2. \exists $p@€€@‚@    E˜< vÎd ?À2DD  DD @DD (?À@DD HDîî Ì 8   @@ @€€À€ ÿ (0%.3.7@PPŸLimit ExistenceLet f be a function and  let c and L be real numbers. The limit of f(x) as x approaches c is %L if and only ifP   €€‚ @  !f„…˜x vc™à Žd !„D !„D ! à„D (!„D@ Hs»€3Îî@Ì @  €@À ‡€€ðà€  (0€@ÿ þ€ (( 0HÀ,5PãInfinite Limits Let f be a fun that is  defined at every real num-ber in some open interval containing c (except &possibly at c itself). The /_statement 2V$d€`¨@   $³ ìð-Ì 0à'$ˆ‚ "$ˆ‚ à"$ˆ‚ P'$ˆ„ ðÝĘ  H@p€PA€ "€'äPD€8_informally A_meansPthat the output of the  function f may be made arbitrarily large by making x sufficiently &close to c.P?Infinite Limit P1 Let c and L be real  numbers and let f and g be functions such that_ V$d€`¨@  $³ ìð-Ì 0à'$ˆ‚ "$ˆ‚ à"$ˆ‚ P'$ˆ„ ðÝĘ  H@p€PA€ "€'äPD€&_and 2W$€ð`@ @ €$³ô‚€-̆$ü€$ˆ„$$ˆ„Dü$ˆƒ„ $ˆ„ÝÇ‚3øb!À€PA€ "€'äPD€/`8`A`P $ p @ €@€ @@€@ €@@ ˜À‚8 8 ãŒ`‡@ a@ aŒ`€ A€ ?à aŒ`€€0C€0 aŒ`€€0?àƒ€0 aŒ`€0€0?à aŒ`€0€0 aŒ`ŠP  P óÞðŒ`  ` @@a€€@@@@€€@ €@€@€à?@ ` ù !À@ThenP   sÀœ`ˆ ˆ œ`sÀ  #PíInfinite Linit P2a Let c and L be real  numbers and let f and g be functions such that_ V$d€`¨@  $³ ìð-Ì 0à'$ˆ‚ "$ˆ‚ à"$ˆ‚ P'$ˆ„ ðÝĘ  H@p€PA€ "€'äPD€%_ 4W$€ð`@ @ €$³ô‚€-̆$ü€$ˆ„$$ˆ„Dü$ˆƒ„ $ˆ„ÝÇ‚3øb!À€PA€ "€'äPD€.}7and@PThen d Z???$ €>`  A4 " A3$³ô‚AØ!4-Ì$b@`!4$ˆDB@@!3$ˆDD@@!4$ˆD 8@ !3$ˆ„@A !3ÝЂ3x#0A4‚† A3‚ €|PA€ "€'äPD€=??d=%dP sÀœ`ˆ ˆ œ` sÀ ", for L > 0P3Infinite Limit P3 Let c and L be real  numbers and let f and g be functions such that_ V$d€`¨@  $³ ìð-Ì 0à'$ˆ‚ "$ˆ‚ à"$ˆ‚ P'$ˆ„ ðÝĘ  H@p€PA€ "€'äPD€%_and 2W$€ð`@ @ €$³ô‚€-̆$ü€$ˆ„$$ˆ„Dü$ˆƒ„ $ˆ„ÝÇ‚3øb!À€PA€ "€'äPD€.`7`@`P ThenX $N @€@€>ƒ°@Ä€À@„€€@ˆ€€@p@@$€‚@@`ðF`€  @€  $³ø-Ì€ð$ˆ€$ˆƒÿÿÿøð$ˆ€$ˆ€ÝÀ @€PA€€ "€=`€'ä €€PD€€€€€!€€ ŒÁ €@@€ X%XP y$ky $yBxB{BBqBB0B$/.PTheorem 1.9b \88v"$H$ð B@` B0  B( "$HH$³pÌ-Ì€$ˆ€$ˆÿÿÿÿÿÿÿÀ$ˆ€$ˆ€ÝÀv€@(@HPD@Ì $@'ä@PD@€ e=eP ,$,BBB~B~B}B~B$~U,,PSTDYø_¼