,limits

    ,:at >e limits1 & :y d we c>e8
  ,"h's a preview 6! solu;n 6"o (!
c5tral pro#ms 9 calculus1 ! pro#m (
-put+ ! slope (a curve4 ,if we want$
6approximate ! slope (! tang5t l9e 6!
curve y .k x^2 at ! po9t
  x .k #1, y .k #1, we cd take ! po9t
(1, 1) al;g )a second po9t on ! curve1
jo9 ^! po9ts )a /rai<t l9e1 & -pute !
slope ( ? secant l9e4 ,= 9/.e1 if !
second po9t 7 (2, 4), !n ! slope (!
secant l9e wd 2 ?4-1_/2-1# .k #3_4 ,if !
second po9t 7 (0, 0), !n ! slope (!
secant l9e wd 2 ?0-1_/0-1# .k #1_4 ,bo?
^! l9es >e %[n 9 ,figure #1, ": x is
cle> t ! f/ is too /eep 6be ! tang5t
l9e1 & t ! second is n /eep 54 ,! slope
(! tang5t l9e m/ "!=e 2 2t #1 & #3_4
  ,6get a bett] approxim,n 6! slope (!
tang5t l9e1 "o "\ 6pick ! second po9t on
! secant l9e 6be clos] to (1, 1)_4 ,=
example1 if ! second po9t 7 (1.5, 2.25),
                                      #j
!n ! slope (! secant l9e wd 2
?2.25-1_/1.5-1# .k #2.5_4 ,if ! second
po9t 7 (0.5, 0.25), !n ! slope (! secant
l9e wd 2 ?0.25-1_/0.5-1# .k #1.5_4 ,ag1
z %[n 9 ,figure #2, ! f/ ( ^! is too
/eep 6be ! tang5t l9e1 &! second is too
%all[4 ,s ! slope (! tang5t l9e m/ 2
"s": 2t #1.5 & #2.5_4
  ,6get a re,y gd approxim,n 6! slope (!
tang5t l9e1 we "\ 6pick ! second po9t
6be v close to (1, 1), s9ce x seems
6make s5se t ! tang5t l9e1 : meets !
curve at only "o po9t1 %d 2 well
approximat$ 0secant l9es 9t]sect+ !
curve at #2 po9ts v close tgr4 ,s we %d
take ! second po9t n at x .k #2 or at
x .k #1.5, b at x .k #1.1 or at
x .k #1.01 or at x .k #1.001_4 ,t is1 we
%d approximate ! tang5t l9e )! l9e "? !
po9t (1, 1) &a po9t l (1.1, 1.1^2") or
(1.01, 1.01^2") or (1.01, 1.01^2")_4 ,!
slopes ( ^! secant l9es >e li/$ 9 !
foll[+ ta#4

                                      #a
      ,'2g9.(c5t].),'

----------------------------------------
;x     (x, f(x))    slope
----------------------------------------
#1.1   (1.1, 1.21)  ?1.21-1_/1.1
                    -1# .k #2.1
----------------------------------------
#1.01  (1.01,       ?1.0201-1_/1.01
        1.0201)     -1# .k #2.01
----------------------------------------
#1.001 (1.001,      ?1.002001-1_/1.001
        1.002002)   -1# .k #2.001
----------------------------------------
#      (1.0001,     ?1.00020001-1_/
1.0001  1.00020001) 1.0001-1# .k #2.0001
----------------------------------------
;x     (x, x^2")    ?x^2"-1_/x-1#
----------------------------------------

      ,'5d.(c5t].),'

  ,x seems cle> t z ;x gets clos] &
clos] to #1, ! slope (! secant l9e gets
                                      #b
clos] & clos] to #2_4 ,x's wor? mak+
sure ? wd 2 true = values ( ;x l #0.999
too2 b x is4 ,s surely ! slope (! tang5t
l9e is #2_4
  ,n[1 :at "o wd re,y l 6d is 6get !
slope (! tang5t l9e xf 0us+ z ! second
po9t (1, 1), s t y g ! slope (a l9e
hitt+ ! curve at only #1, po9t4 ,b if y
try ?1 ! slope y -pute is
?1-1_/1-1# .k ?0_/0#, : is n a numb]4
  ,? is wor? /ress+ pretty firmly3 ?0_/
0# is n a numb]4 ,x's n #0, ev5 ?\< #0
ov] any?+ except #0 is #0_4 ,x's n +-,=,
ev5 ?\< any?+ b #0 mi<t seem 6give +-,=
:5 divid$ by #0_4 ,x's n #1, ev5 ?\<
any?+ b #0 divid$ 0xf is #0_4 ,x's n a
numb]4 ,9 fact1 y wdn't want ?0_/0# 6be
#0 or #1 or +-,= 9 ? case---y'd want x
6be #2_4 ,s x's j z well t---,:at did we
say8 ?0_/0# is n a numb]4
  ,algebraic,y1 !n1 ! /ory is ?4
,6approximate ! slope (! tang5t l9e to
y .k x^2 at ! po9t (1, 1), we take a
second po9t (x, x^2") close to (1, 1) on
                                      #c
! curve1 & -pute ! slope (! secant l9e
jo9+ (1, 1) & (x, x^2")_4 ,? slope is
  s(x) .k ?x^2"-1_/x-1#_4
  ,:5 x .k #1, ? slope is undef9$4 ,b :5
;x is close to #1, ! slope "\ 6be close
6! slope (! tang5t l9e4 ,s :at we want
6be a# 6-pute is ! value t s(x) gets
close 6:5 ;x is close to #1 b x /.k #1_4
,? numb] is call$ ! .limit ( s(x) z ;x
approa*es #1_4 ,x is writt5
  "lim%x $o #1] s(x)
    .k "lim%x $o #1] ?x^2"-1_/x-1#_4
  ,9 ? case1 ! ta# abv makes a /r;g case
t ! value ( ? limit %old 2
  "lim%x $o #1] s(x)
    .k "lim%x $o #1] ?x^2"-1_/x-1#
    .k #2_4
  ,? is :at limits >e1 & ? is :y we c>e
ab !m3 we ne$ !m 9 ord] 6-pute ! slopes
( curves4 ,we'll see lat] t we al ne$
limits 9 ord] 6-pute >1s ( -plicat$
figures1 : we'll describe z limits (!
>1s ( simple figures (collec;ns (
rectangles) t approximate ^? -plicat$
                                      #d
figures4

    ,pictures ( func;ns ) limits
  ,an 9tuitive way ( ?9k+ ab limits is
6say t "lim%x $o a] f(x) is ! value y
expect f(a) 6take z l;g z ;f doesn't d
any?+ funny at ! po9t x .k a_4 ,"! >e #3
pictures wor? keep+ 9 m9d :5 "o ?9ks ab
func;ns hav+ limits4 ,! f/1 ,figure #3,
%[s a normal1 well-2hav$ func;n ;f go+
smoo?ly "? ! po9t (2, 1)_4 ,:5 ;x is
close to #2 b x /.k #2, f(x) is close to
#1_2 s "lim%x $o #2] f(x) .k #1_4 ,al1
f(2) .k #1_4
  ,! second figure1 ,figure #4, %[s a
func;n ;g t is nice & well-2hav$ "ey":
except at x .k #3_4 ,at ? "o po9t1 !
func;n ;g is undef9$4 ,tradi;n,y1 "o
repres5ts ? ) an op5 circle 9 ! graph4
,despite ! hole1 ?\<1 x is / ! case t :5
;x is close to #2 b x /.k #2, f(x) is
close to #1_2 s "lim%x $o #2] g(x) .k #1
, ev5 ?\< g(2) xf isn't def9$4

                                      #e
  ,! la/ figure1 ,figure #5 %[s an ev5 m
pa?ological func;n1 ;h_4 ,! graph (
y .k h(x) has a hole at ! po9t (2, 1), b
!n1 unexpect$ly1 h(2) .k #2_4 ,x is / !
case t :5 ;x is close to #2 b x /.k #2,
h(x) is close to #1_2 s
"lim%x $o #2] h(x) .k #1, ev5 ?\<
h(2) .k #2_4
  ,x wd 2 easy 6imag9e t ! only "o ( ^!
#3 situ,ns t matt]s 0 ! "o 9 ,figure #3
)! normal func;n1 & t ! o!r two figures
>e j obnoxi\s pa?ologies2 b ? isn't "r4
,x is true t ! situ,n 9 ,figure #5, ": !
func;n ;h is def9$ at x .k #2 b ": xs
value "! is unexpect$1 is an obnoxi\s
pa?ology---a 3triv$ func;n ( no import.e
:atso"e4 ,if we only d1lt ) 3t9u\s
func;ns l ! "o 9 ,figure #3 ?\<1 !n we
wdn't ne$ ! idea ( "lim%x $o a] f(x) at
all4 ,we cd j use ! simpl] idea ( f(a)_4
  ,! situ,n t re,y matt]s 6u is ! "o %[n
9 ,figure #4, ": ! func;n ;g is undef9$
at ! v po9t t matt]s 6u4 ,af all1 !
limit t >ose 9 ! slope pro#m abv 0
                                      #f
  "lim%x $o #1] s(x)
    .k "lim%x $o #1] ?x^2"-1_/x-1#_4
  ,:at wd ? look l graphic,y8 ,! func;n
s(x) .k ?x^2"-1_/x-1# looks l a /rai<t
l9e1 except t x is undef9$ at ! po9t
(1, 2), ": ! denom9ator is #0_4 ,figure
#6 %[s ?4
  ,? isn't surpris+1 s9ce z l;g z
x /.k #1, we h
  s(x) .k ?x^2"-1_/x-1#
    .k ?(x-1)(x+1)_/x-1# .k x+1,
s t ! graph ( s(x) .k ?x^2"-1_/x-1# %d
look j l ! graph ( y .k x+1, except t x
%d 2 undef9$ at ! po9t (1, 1)_4
,annoy+ly1 ! "o po9t ": s(x) is undef9$
is ! "o po9t ": we c>e ab xs value4
  ,? is re,y :at limits >e =3 plu7+ up
s+le po9ts ": func;ns annoy+ly manage
6be undef9$4

    ,pictures ( func;ns )\t limits
  ,figures #7, #8, & #9 %[ #3 ways =a
func;n n 6h a limit at a po9t4

                                      #g
  ,figure #7 %[s a func;n ;f = :
"lim%x $o #1] f(x) does n exi/4 ,"h !
pro#m is obvi\s4 ,! func;n ;f approa*es
di6]5t limits z x $o #1 f ! left & z
x $o #1 f ! "r2 s "! is no "o limit =!
func;n t applies reg>d.s ( direc;n4
  ,figure #8 al %[s a func;n : has no
limit z x $o #1, & "h ag1 ! pro#m is
obvi\s3 ! func;n has a v]tical asymptote
at ! po9t x .k #1_2 s x _c h a limit at
? po9t4
  ,figure #9, : %[s a func;n h(x) ) no
limit z ;x approa*es #0, is m subtle4 ,!
func;n "h is h(x) .k sin (1_/x)_4 ,?
func;n has 9f9itely _m p1ks & valleys :
get clos] & clos] tgr z ;x approa*es #0
f ei side4 ,"! is "!=e n any "o numb] t
h(x) approa*es z x $o #0_4 ,maple writes
! limit 9 ? case z -#1..1, m1n+ t e
numb] 2t -#1 & #1 is a limit po9t (!
func;n4 ,? is non-/&>d1 b 9=mative4
,/&>d ma!matical usage wd j 2 6say t !
limit does n exi/4

                                      #h
     ,a =mal def9i;n ( limits
  ,! 9=mal def9i;n (
"lim%x $o a] f(x) .k ,l is t :5 ;x is
close to ;a b x /.k a, !n f(x) is close
to ;,l_4 ,m precisely1 we c say t we c
make f(x) z close to ;,l z we l1 z l;g z
we make ;x close 5 to ;a_4 ,? is ! 3t5t
(! (ficial def9i;n (
"lim%x $o a] f(x) .k ,l, : says t x m1ns
t giv5 any .@e .1 #0, we c f9d a
.d .1 #0 s* t \f(x)-,l\ "k .@e :5"e
#0 "k \x-a\ "k .d_4 ,9 o!r ^ws1 no matt]
h[ close1 .@e, y want 6make f(x) to ,l,
y c alw get x t close z l;g z y pick ;x
close 5 to a, at mo/ a 4t.e .d away4 ,!
geometric idea is %[n 9 ,figure #10_4
,no matt] h[ small ! 4t.e .@e is *os5
6be1 "o has 6be a# 6f9d a 4t.e .d s* t e
po9t )9 a 4t.e .d ( ;a (except = maybe
;a xf) is mapp$ by ;f 96! 9t]val
,l-.@e "k y "k ,l+.@e_4

    ,simple algebraic -put,ns ( limits4

                                      #i
  ,we'll ne$ 6sp5d "s "t "w+ \ me?ods =
-put+ limits1 b "! >e a few ?+s we c say
"r away4
  ,"! >e simple rules = -put+ ! limits
we don't c>e ab1 : >e (t5 2labor$
0calculus books4 ,^! rules basic,y say t
! value (! limit (a simple func;n )\t
holes 9 xs graph is ! value (! func;n at
t po9t4
  ,if ;c is a 3/ant1 !n
"lim%x $o a] c .k c_4 (,no surprise1 "r8
,:5 ;x gets close to a, ;c gets close to
;c_4)
  "lim%x $o a] x .k a_4 (,and :5 ;x gets
close to a, ;x gets close to1 let's see
''' ;a_4)
  ,if "lim%x $o a] f(x) .k ,l &
"lim%x $o a] g(x) .k ,m exi/1 !n
  "lim%x $o a] (f(x)+g(x))
    .k ("lim%x $o a] f(x))
    +("lim%x $o a] g(x)) .k ,l+,m_4
  ,! same rule "ws = f(x)-g(x) &=
f(x)g(x)_4 ,x al "ws = f(x)_/g(x) z l;g
z ,m /.k #0_4
                                     #aj
  ,>m$ ) ^! rules1 we c d limits l
  "lim%x $o #2] ?x^2"+5_/x-1#
    .k ?"lim%x $o #2] (x^2"+5)_/
      "lim%x $o #2] (x-1)#
    .k ?"lim%x $o #2] (x^2")+
      "lim%x $o #2] #5_/"lim%x $o #2] x-
      "lim%x $o #2] #1#
    .k ?("lim%x $o #2] x)("lim%x $o #2]
      x)+"lim%x $o #2] #5_/"lim%x $o #2]
      x-"lim%x $o #2] #1#
    .k ?2*2+5_/2-1#
    .k ?2^2"+5_/2-1# .k #9_4
  ,^! limits don't 9t]e/ u v m*4
  ,! m important obs]v,n is t *ang+ !
value (a func;n at "o po9t doesn't *ange
! value ( any limit 9volv+ t func;n4 ,if
y >e -put+ ! limit z x $o a, !n *ang+ !
value ( f(a) x makes no di6];e1 2c !
limit dep5ds only on :at happ5s :5 ;x is
close to ;a b x /.k a_4 ,*ang+ ! value (
f(b) = any b /.k a doesn't *ange !
limit1 s9ce ! limit dep5ds (! value (
f(x) = ;x v close to a, clos] ?an any
fix$ ;b_4 ,we c "!=e d ?+s l ?3
                                     #aa
  ,! func;ns s(x) .k ?x^2"-1_/x-1#
.k ?(x-1)(x+1)_/x-1# & t(x) .k x+1 di6]
only at "o po9t3 s(1) is undef9$1 :ile
t(1) .k #2_4 ,x m/ "!=e 2 ! case t s(x)
& t(x) h ! same limit at e po9t2 s
  "lim%x $o #1] ?x^2"-1_/x-1#
    .k "lim%x $o #1] (x+1)
    .k "lim%x $o #1] x+"lim%x $o #1] #1
    .k #1+1 .k #2_4
  ,at ? /age1 we h f9,y -put$ ! slope (!
tang5t l9e to y .k x^2 at ! po9t x .k #1
9 -plete detail1 & we c 2 c]ta9 (!
value1 #2, t we 3jectur$ at ! 2g9n+ ( \r
4cus.n ( limits4
  ,n[ mi<t 2 a gd mo;t 6rer1d ! 4cus.n (
slopes at ! 2g9n+ ( ? sec;n1 & 6make
sure x makes s5se4

    ,algebraic -put,n ( m -plicat$
    limits

,review4
  ,at ? /age1 we h #2 pr9ciples = -put+
"lim%x $o a] f(x)_4
                                     #ab
  (1) ,-pute f(a)_4 ,if y get a m1n+;l
value1 & if is ;f is nice & 3t9u\s1 !n
"lim%x $o a] f(x) .k f(a)_4
  (2) ,if :5 y try 6-pute f(a), y get
?0_/0#, !n look =& remove -mon factors 9
! num]ator & denom9ator ( f(x), & try
ag4
  ,an example (! f/ process wd 2
  "lim%x $o #3] ?x^2"-2x+5_/3x-1#
    .k ?3^2"-2(3)+5_/3(3)-1# .k ?8_/8#
    .k #1_4
,! details "h cd 2 ju/ifi$ 0\r !orems ab
! limit (a sum1 di6];e1 product1 &
quoti5t1 & ab ! limit (a 3/ant &! limit
( ;x_4
  ,an example (! second process wd 2
  "lim%x $o #2] ?x^3"-8_/2x-4#_4
,if we try j plu7+ 9 x .k #2, we get
?2^2"-8_/2(2)-4# .k ?0_/0#, : is
undef9$4 ,we "!=e try 6f9d & remove -mon
factors f ! num]ator & denom9ator4 ,s9ce
! denom9ator is #2(x-2), we ev5 "k :at
-mon factor we want 6f9d & elim9ate3
x-2_4 ,we cd "!=e ei rememb] h[ !
                                     #ac
num]ator factors1 or we cd divide !
num]ator by x-2 & see :at em]ges4 ,we wd
f9d t x^3"-8 .k (x-2)(x^2"+2x+4), s t !
limit 2comes
  "lim%x $o #2] ?x^3"-8_/2x-4#
    .k "lim%x $o #2] ?(x-2)(x^2"+2x
    +4)_/2(x-2)#
    .k "lim%x $o #2] ?x^2"+2x+4_/2#
    .k ?2^2"+2(2)+4_/2# .k #6_4
  ,x's wor? ?9k+ a mo;t ab :at has
happ5$ "h 3ceptu,y4 ,:5 we c.el ! -mon
factor ( x-2 6move f ?(x-2)(x^2"+2x+4)_/
2(x-2)# to ?x^2"+2x+4_/2#, we don't q
get ! same func;n back1 s9ce ?x^2"+2x+4
_/2# is def9$ = e r1l numb] ;x, :ile
?(x-2)(x^2"+2x+4)_/2(x-2)# is undef9$ at
x .k #2_4 ,except at ? "o "o po9t1 h["e1
! two func;ns agree "ey":4 ,we alr saw1
?\<1 t *ang+ ! value (a func;n at "o
po9t does n *ange any limit 9volv+ t
func;n2 s t ! calcul,n abv makes s5se4

,ano!r example ( elim9at+ -mon factors4

                                     #ad
  ,"s"ts y j ne$ 6d algebra 6remove -mon
factors 9 ! num]ator & denom9ator4
,3sid]1 = 9/.e1
  "lim%x $o #3] ,??1_/x#-?1_/3#,_/x
    -3,#_4
,alm alw1 ,i wd />t s* a pro#m 0rewrit+
! frac;ns 6be on #2 levels r ?an #3_4
,x's too h>d 6?9k ab o!rwise4 ,y get
  "lim%x $o #3] (?1_/x-3#@(?1_/x#
    -?1_/3#@))_4
,! only ?+ t -es 6m9d is 6put "ey?+ ov]
a -mon denom9ator & 6hope =! be/4 ,y get
  "lim%x $o #3] (?1_/x-3#@(?3_/3x#-?x_/
      3x#@))
    .k "lim%x $o #3] (?1_/x-3#@(?3-x_/
      3x#@))
    .k "lim%x $o #3] (-?1_/3x#) .k -?1_/
      9#_4

,! trick ) squ>e roots4
  ,3sid]
  "lim%x $o #3] ?>x]->3]_/x-3#_4
,if we />t \ & plug 9 x .k #3, we get
?0_/0#_4 ,b h[ c we remove a -mon factor
                                     #ae
f ! num]ator & denom9ator8 ,"o /&>d
answ] is 6multiply by #1 9 a -plicat$
way3
  "lim%x $o #3] ?>x]->3]_/x-3#
    .k "lim%x $o #3] (?>x]->3]_/x-3#*
      ?>x]+>3]_/>x]+>3]#)
    .k "lim%x $o #3] ?x-3_/(x-3)(>x]+>3]
      )#
    .k "lim%x $o #3] ?1_/(>x]+>3])# .k ?
      1_/>3]+>3]# .k ?1_/2>3]#_4

,ano!r view (! trick ) squ>e roots4
  ,"s p f9d x simpl] 6?9k (! example abv
l ?3 ,! denom9ator1 x-3, is actu,y !
di6];e ( two squ>es1
  x-3 .k (>x])^2"-(>3])^2_4
,x c "!=e 2 factor$ z
x-3 .k (>x]->3])(>x]+>3])_4 ,\r limit is
"!=e
  "lim%x $o #3] ?>x]->3]_/x-3#
    .k "lim%x $o #3] ?>x]->3]_/(>x]
    ->3])(>x]+>3])#
    .k "lim%x $o #3] ?1_/>x]+>3]#
    .k ?1_/>3]+>3]# .k ?1_/2>3]#_4
                                     #af
  ,use :i*"e ( ^! tricks seems mo/
natural 6y4

,! ,squeeze ,!orem4
  ,if "! >e #3 func;ns1 ;f, ;g, ;h def9$
ne> ;a, & if at e po9t except maybe at
;a we h f(x) "k g(x) "k h(x), & if
  "lim%x $o a] f(x)
    .k "lim%x $o a] h(x) .k ,l,
!n "lim%x $o a] g(x) .k ,l z well4
  ,9 ! normal applic,ns ( ? !orem1 ;f &
;h >e simple func;ns ^: limits we c
-pute1 & ;g is a -plicat$ func;n we c't
analyze directly4 ,figure #11 %[s !
idea4
  ,6see t ! ,squeeze ,!orem is "s"ts
nec1 3sid] ! func;n
f(x) .k xsin (?1_/x#), %[n 9 ,figure
#12_4
  ,we wd l 6be a# 6say t s9ce
"lim%x $o #0] x .k #0, x foll[s t
  "lim%x $o #0] (xsin (?1_/x#))
    .k ("lim%x $o #0] x)
    ("lim%x $o #0] sin (?1_/x#))
                                     #ag
    .k #0*("lim%x $o #0] sin (?1_/x#))
    .k #0_4
,? isn't valid1 h["e1 s9ce we've alr se5
t "lim%x $o #0] sin (?1_/x#) doesn't
exi/4 ,:ile any numb] "ts #0 is gu>ante$
6be #0, we don't "k any?+ ab :at !
product ( #0 & "s non-exi/5t ?+ mi<t 24
,= 9/.e1 x wd cle>ly 2 9valid 6say
  "lim%x $o #0] x?1_/x#
    .k ("lim%x $o #0] x)
    ("lim%x $o #0] ?1_/x#)
    .k #0*("lim%x $o #0] ?1_/x#) .k #0,
s9ce "lim%x $o #0] x?1_/x#
.k "lim%x $o #0] #1 .k #1_4
  ,! ticket 6-put+
"lim%x $o #0] (xsin (?1_/x#)) is !
,squeeze ,!orem4 ,we "k t z l;g z
x /.k #0,
  -#1 "k: sin (1_/x) "k: #1
  -\x\ "k xsin (1_/x) "k \x\_4
,x is al cle> t
  "lim%x $o #0] (-\x\) .k #0
    .k "lim%x $o #0] \x\,
s 0! ,squeeze ,!orem1
                                     #ah
  "lim%x $o #0] (xsin (?1_/x#)) .k #0_4

,two tricky trig limits4
  ,plott+ po9ts or mak+ a ta# ( values
makes x appe> t
  "lim%x $o #0] ?sin x_/x# .k #1
& t
  "lim%x $o #0] ?1-cos x_/x^2"#
    .k ?1_/2#_4
,lat]1 we'll ne$ ^! limits =a c\ple (
te*nical calcul,ns4 ,n[1 let's see :y !y
>e true4 ,if y plug 9 x .k #0 6! f/
func;n1 y get ?sin #0_/0# .k ?0_/0#, an
undef9$ result4 ,\r g5]al approa* wd say
6remove a -mon factor 2t ! num]ator &
denom9ator4 ,b h[ c we factor ;x \ f
sin x_8 ,"! is no obvi\s way4
  ,! trick turns \ 6be a cl"e use (!
,squeeze ,!orem1 al;g ) "s sm>t
geometry4 ,f/ ( all1 ?9k ab \ /&>d draw+
(a po9t on ! unit circle1 9 ,figure #13
_4
  ,if .? is ! angle %[n1 !n ! l5g? (! >c
m>k$ \ 0? angle is al .?_4 ,! l5g? (!
                                     #ai
v]tical side (! triangle %[n is sin .?_4
,x is cle> geometric,y t ! >c is l;g]
?an ! side (! triangle2 s sin .? "k .?,
or ?sin .?_/.?# "k #1_4
  ,n[ look at a less obvi\s bit (
geometry4 ,figure #14 %[s ! same angle
.? al;g )a triangle =m$ 0a v]tical l9e
"? (1, 0)_4
  ,! >ea (! :ole unit circle ( #2.p
radians is .p, &! >ea (! 8pie slice0
m>k$ 0angle .? m/ 2 ! >ea (! :ole
circle1 .p, multipl$ 0! frac;n (! :ole
circle t lies 9side ! pie slice1 ?.?_/
2.p#_4 ,9 o!r ^ws1 ! >ea (! pie slice is
.p(?.?_/2.p#) .k ?.?_/2#_4 ,s9ce
tan .? .k ?opp_/adj#, ! v]tical side (!
triangle m/ 2 tan .?_4 (,9cid5t,y1 ? is
": ! tang5t func;n got xs "n3 x's ! l5g?
(! v]tical seg;t t is tang5t 6! circle
_4) ,! >ea (! triangle is "!=e
  ?1_/2#bh .k ?1_/2#*1*tan .?
    .k ?tan .?_/2#_4
  ,x is cle> geometric,y t ! triangle is
bi7] ?an ! 8pie slice10 s x m/ 2 t
                                     #bj
  ?.?_/2# "k ?tan .?_/2#
  .? "k tan .? .k ?sin .?_/cos .?#
  ?sin .?_/.?# .1 cos .?_4
  ,putt+ ? tgr )! previ\s 9equal;y gives
  cos .? "k ?sin .?_/.?# "k #1_4
  ,s9ce "lim%.? $o #0] cos .?
.k "lim%.? $o #0] #1 .k #1, x foll[s f !
,squeeze ,!orem t
  "lim%.? $o #0] ?sin .?_/.?# .k #1_4
  ,? 0 ! f/ (! two cl"e trig limits4 ,!
second c 2 gott5 algebraic,y f ! f/4 ,we
use a trick l ! squ>e root trick 6obs]ve
t
  ?1-cos .?_/.?^2"#
    .k ?1-cos .?_/.?^2"#*?1+cos .?_/1
    +cos .?#
    .k ?1-cos^2 .?_/.?^2"(1+cos .?)#
    .k ?sin^2 .?_/.?^2"(1+cos .?)#_4
,? m1ns t
  "lim%.? $o #0] ?1-cos .?_/.?^2"#
    .k "lim%.? $o #0] ?sin^2 .?_/.?^2"(1
      +cos .?)#
    .k ("lim%.? $o #0] ?sin .?_/.?#)(
      "lim%.? $o #0] ?sin .?_/.?#)(
                                     #ba
      "lim%.? $o #0] ?1_/1+cos .?#)
    .k #1*1*?1_/2# .k ?1_/2#,
: is ! second cl"e trig limit4 ,x is
wor? /at+ a coroll>y1 : is ! result we
actu,y ne$1
  "lim%.? $o #0] ?1-cos .?_/.?#
    .k "lim%.? $o #0] (?1-cos .?_/.?^2"#
      *.?)
    .k ("lim%.? $o #0] ?1-cos .?
      _/.?^2"#)("lim%.? $o #0] .?)
    .k ?1_/2#*0 .k #0_4

    ,3t9u;y2 ,"o-sid$ & ,9f9ite ,limits

,3t9u;y at a po9t4
  ,9tuitively1 a func;n is 3t9u\s if y c
draw ! graph )\t pick+ up yr p5cil4 ,!
idea ( limits lets u take ? vague idea &
r5d] x precise4 ,"h's ! (ficial def9i;n3
  ,! func;n ;f is 3t9u\s at ! po9t
x .k a if & only if
"lim%x $o a] f(x) .k f(a)_4 ,? m1ns #3
?+s3

                                     #bb
  (1) "lim%x $o a] f(x) has 6exi/4
  (2) f(a) has 6exi/4
  (3) "lim%x $o a] f(x) & f(a) h 6be
equal4
  ,"! >e "!=e #3 ways a func;n cd g ab n
2+ 3t9u\s at ! po9t x .k a_3 any (! #3
3di;ns 9 ! def9i;n cd 2 false4
  ,let's try 63v9ce \rvs t ! def9i;n "h
captures ! idea ( 3t9u;y4 ,f/ ( all1 x's
easy 6see t normal func;ns at normal
po9ts >e 3t9u\s4 ,= 9/.e1 if
f(x) .k x^2, !n ;f is 3t9u\s at x .k #2
(or at any o!r po9t1 = t matt]) 2c
  "lim%x $o #2] f(x)
    .k "lim%x $o #2] x^2 .k #4 .k #2^2
    .k f(2)_4
,9 g5]al1 if y c -pute "lim%x $o a] f(x)
0j plu7+ x .k a 9to f(x), !n ;f is
3t9u\s at ;a_4

,4cont9u\s func;ns4
  ,s9ce s _m normal func;ns >e 3t9u\s
"ey":1 we may ga9 m 9si<t 96:at 3t9u;y
m1ns 0/udy+ ! 4cont9u\s func;ns4 ,t is1
                                     #bc
we'll try 6le>n ab :at 3t9u;y is 0le>n+
ab :at 3t9u;y isn't4 ,a func;n is
4cont9u\s at x .k a if x fails 6satisfy
"o or m (! 3di;ns abv4 ,we h 6look a bit
6f9d s* func;ns1 b let's locate a bun* (
^! 4cont9u\s func;ns & see t !y al fail
6satisfy \r 9tuitive id1s ab 3t9u;y4

  ..,func;ns 4cont9u\s 2c "lim%x $o a] f
(x) does n .exi/4

  ,! func;ns pictur$ 9 ,figures #7, #8,
#9 fail 6be 3t9u\s at ! po9ts at : !y d
n h limits4 ,t is1 a func;n )a br1k at
x .k a ": x approa*es di6]5t limits f !
left &! "r fails 6be 3t9u\s at x .k a_4
,simil>ly1 a func;n )a v]tical asymptote
at x .k a fails 6be 3t9u\s at x .k a, &a
func;n ) 9f9itely _m waves (! same hei<t
9 e nei<borhood ( x .k a fails 6be
3t9u\s at ;a_4



                                     #bd
  ..,func;ns 4cont9u\s 2c f(a) does n
.exi/4

  ,! func;n pictur$ 9 ,figure #4, : is a
smoo? func;n except t x is undef9$ at "o
po9t1 al fails 6be 3t9u\s at ! po9t
x .k #2 ": x is undef9$4 ,al?
"lim%x $o #3] f(x) .k #1, f(2) is
undef9$1 & _c equal ! limit4

  .,func;ns .": "lim%x $o a] f(x) /.k f
(a)_4

  ,f9,y1 = -plete;s1 ! func;n 9 ,figure
#5, : approa*es #1 z x $o #2, b : !n has
a jump at ? "o po9t s t "! is a hole 9 !
graph at (2, 1) &a s+le dot at (2, 2),
is 4cont9u\s at x .k #2, s9ce ev5 ?\<
bo? ! limit &! func;n >e def9$ at
x .k #2, we h
  "lim%x $o #2] f(x) .k #1 /.k #2
    .k f(2)_4
  ,all ( ^! func;ns seem l !y %d 2
4cont9u\s at ^! po9ts us+ \r 9tuitive
                                     #be
no;n ( 4cont9u\s func;ns z "os ": y h
6pick up yr p5cil 6draw ! graph2 s !
=mal def9i;n ( 3t9u\s func;ns seems l a
gd "o4

,3t9u;y 9 an 9t]val4
  ,! def9i;n (a 3t9u\s func;n talk$ ab
3t9u;y at "o po9t4 ,:at we >e m (t5
9t]e/$ 9 is 3t9u;y 9 an 9t]val4 ,a
func;n ;f is 3t9u\s on an 9t]val if &
only if x is 3t9u\s at e po9t 9 !
9t]val4 ,x is 3t9u\s1 p]iod1 if x is
3t9u\s "ey":4

,easy ,!orems on ,3t9u;y4
  ,! !orems ab ! sum1 product1 di6];e1 &
quoti5t ( limits 2+ ! limits (! sum1
product1 di6];e1 & quoti5ts (! func;ns
9volv$ give rise 6easy !orems say+ t !
sum1 product1 di6];e1 & quoti5t ( #2
3t9u\s func;ns >e !mvs 3t9u\s4 ,= 9/.e1
if ;f & ;g >e 3t9u\s at ;a, !n
  "lim%x $o a] (f(x)g(x))
    .k "lim%x $o a] f(x)
                                     #bf
    *"lim%x $o a] g(x) .k f(a)g(a),
s fg is 3t9u\s at ;a_4 ,0! same r1son+1
f+g & f-g >e 3t9u\s at a, z is f_/g if
g(a) /.k #0_4

,9t]m$iate ,value ,!orem4
  ,if ;f is 3t9u\s on a clos$ 9t]val
@(a, b@), & if ;k is any numb] 2t f(a) &
f(b), !n "! is "s po9t ;c, a "k c "k b,
= : f(c) .k k_4
  ,= 9/.e1 ,figure #15 %[s a func;n
f(x) .k (x-2)^3"-2(x-2)+4_4 ,? func;n is
a polynomial1 s x is 3t9u\s at e po9t on
@(0, #4@)_4 ,at ! 5dpo9ts1 f(0) .k #0 &
f(4) .k #8_4 ,if we let k .k #4, !n
f(0) .k #0 "k k .k #4 "k f(4) .k #8_2 s
0! ,,ivt1 "s po9t ;c c 2 f.d at :
f(c) .k #4_4 ,9 fact1 x appe>s t "! >e
#3 s* po9ts4
  ,on an 9tuitive level1 ! ,,ivt is an
obvi\s fact4 ,if y />t \ on "o side (
y .k k & y draw at curve )\t pick+ up yr
p5cil1 & y 5d up on ! o!r side ( y .k k,
!n at "s po9t 9 2t1 y m/ h cross$
                                     #bg
y .k k_4 ,x turns \1 ?\<1 t x is n at
all trivial 6give a =mal pro( ( ? fact f
! usual axioms =! r1l numb]s1 & us+ !
=mal def9i;n ( 3t9u;y 9/1d ( \r 9=mal
no;n4 ,9 ! book ,i le>n$ calculus f1 !
,,ivt 0 9 a *apt] refre%+ly 5titl$1 8
,?ree ,h>d ,!orems40

    ,ext5.ns (! ,limit ,3cept

,"o-sid$ limits4
  ,let f(x) 2 ! func;n %[n abv 9 ,figure
#7_4 ,!n "lim%x $o #1] f(x) does n exi/1
s9ce "! is s+le numb] t ! values ( f(x)
>e approa*+ z ;x approa*es #1_4 ,on !
o!r h&1 if ;x is close to #1 & x "k #1,
!n ! graph %[s t f(x) is close to #2_4
,we describe ? situ,n1 t f(x) is close
to #2 :5"e ;x is close to #1 & x "k #1,
0writ+
  "lim%x $o #1^-]  f(x) .k #2_4
,simil>ly1 ! graph %[s t f(x) is close
to #1 :5"e ;x is close to #1 & x .1 #1_4
,we write ? z
                                     #bh
  "lim%x $o #1^+]  f(x) .k #1_4
,^! two quantities1
"lim%x $o #1^-]  f(x) &
"lim%x $o #1^+]  f(x), >e call$ ! limit
( f(x) z ;x approa*es ;a f ! left (or f
2l) &! limit ( f(x) z ;x approa*es ;a f
! "r (or f abv)_4 ,"!'s no v subtle idea
"h---x's j not,n4 ,obvi\sly if
  "lim%x $o a^-]  f(x)
    .k "lim%x $o a^+]  f(x) .k ,l,
!n f(x) gets close to ;,l :5"e ;x gets
close to a, :e!r f ! "r or f ! left2 s t
  "lim%x $o a] f(x) .k ,l
z well4 ,equ,y obvi\sly1 if
  "lim%x $o a] f(x) .k ,l,
!n
  "lim%x $o a^-]  f(x)
    .k "lim%x $o a^+]  f(x) .k ,l_4
  ,x's wor? 2+ explicit t ! limit z ;x
approa*es ;a f 2l m1ns ! limit z ;x
approa*es ;a f small] values ( x, n (
;y_4 ,9 ! func;n 9 ,figure #7, lim
;x ;$o #1;^-"f(x) is #2, n #1_4

                                     #bi
  ,x's al wor? not+ t m obnoxi\s func;ns
l ! "o 9 ,figure #9 abv may lack "o-sid$
limits3 n"o (! limits
  "lim%x $o #0] sin (?1_/x#),
    ="lim%x $o #0^+]  sin (?1_/x#),
    ="lim%x $o #0^-]  sin (?1_/x#)
exi/4

,v]tical ,asymptotes & ,9f9ite ,limits4
  ,we've s f> look$ a lot at -put+
limits ( func;ns t look l
  "lim%x $o c] f(x)
    .k "lim%x $o c] ?a(x)_/b(x)#_4
,\r basic rules h be5 6try 6j plug 9
x .k c & get a(c)_/b(c)_4 ,if ? is a
numb]1 & if ! num]ator & denom9ator >e
3t9u\s func;ns (polynomials1 = example)1
!n
  "lim%x $o c] ?a(x)_/b(x)#
    .k ?a(c)_/b(c)#,
& we're d"o4
  ,if1 on ! o!r h&1 we plug 9 x .k c &
we get ?a(c)_/b(c)# .k ?0_/0#, : is n a
numb]1 !n we h a :ole rep]toire ( tricks
                                     #cj
6f9d & elim9ate -mon factors 2t !
num]ator & denom9ator1 af : we plug 9
x .k c ag1 & see :at happ5s4
  ,b :at if we plug 9 x .k c & we get n
?a(c)_/b(c)# .k ?2_/3# or
?a(c)_/b(c)# .k ?0_/0#, b
?a(c)_/b(c)# .k ?3_/0#, or1 9 g5]al1
?a(c)_/b(c)# .k ?p_/0#, ": ;p is a
nonz]o numb]8 ,assume bo? ! num]ator &
denom9ator >e 3t9u\s at ;c_4 ,!n :5 ;x
is a numb] close to ;c, b x /.k c, a(x)
w 2 close to ;p & b(x) w 2 close to #0_4
,! result w 2 t ?a(x)_/b(x)# w h a
normal siz$ num]ator &a t9y denom9ator1
s t ?a(x)_/b(x)# w ei 2 huge & positive
(l #10^100")1 or huge & negative (l
-#10^100")_4 ,! clos] y take ;x to ;c, !
clos] ! denom9ator w get to #0, &! bi7]
! frac;n w get 9 absolute value4 ,9 o!r
^ws1 ?a(x)_/b(x)# w h a v]tical
asymptote at x .k c_4
  ,"! >e two possi# ?+s "o cd "!=e say
ab "lim%x $o c] ?a(x)_/b(x)#_4 ,"o
p]fectly h"o/ analysis wd simply 2 6say
                                     #ca
t ! limit does n exi/4 ,? is absolutely
correct bas$ on ! prop] def9i;n ( limits
9 t]ms ( .@e & .d_4 ,b x's re,y a lot m
9=mative 6try 6det]m9e :e!r ?a(x)_/b(x)#
is big & positive or big & negative on
ea* side ( x .k c_4 ,>e we1 = 9/.e1 9 a
situ,n l f(x) .k ?1_/x# 9 ,figure #16,
":1 )! obvi\s m1n+ =! symbols1
  "lim%x $o #0^+]  ?1_/x# .k +,=
    &  "lim%x $o #0^-]  ?1_/x# .k -,=,
  or >e we 9 a situ,n l
f(x) .k ?1_/x^2"# 9 ,figure #17, ":
  "lim%x $o #0^+]  ?1_/x^2"# .k +,=
    &  "lim%x $o #0^-]  ?1_/x^2"#
    .k +,=_4
  ,norm,y1 ! way 6figure \ : is ! case
is 6plug 9 numb]s close to ;c, ei
num]ic,y or 3ceptu,y4 ,t is1 y cd guess
t
  "lim%x $o #0^-]  ?1_/x# .k -,=
2c ?1_/0.0001# .k -#10000, or y cd say t
3ceptu,y1 9 -put+ ! limit y >e do+ "s?+
l
  ?1_/-tiny# .k ,huge_4
                                     #cb
  ,6pick a m -plex example1 suppose we
want$ 6-pute
  "lim%x $o #2^+]  ?x^3"-7x+11_/4x^2"
    -20x+24#_4
,we wd />t 0plu7+ 9 x .k #2, : wd make !
frac;n 9to ?5_/0#_4 ,? wd tell u t we h
a v]tical asymptote at x .k #2_4 ,"k+ t
! denom9ator is #0 :5 x .k #2, we wd
natur,y factor (x-2) \ (! denom9ator
6get
  "lim%x $o #2^+]  ?x^3"-7x+11_/4(x-2)(x
    -3)#_4
,if ;x is sli<tly l>g] ?an #2, !n !
frac;n is r\<ly
  ?5_/4(+tiny)(-1)#
    .k -?5_/4#*?1_/+tiny#
    .k -?5_/4#(+,huge) .k -,huge,
s
  "lim%x $o #2^+]  ?x^3"-7x+11_/4x^2"
    -20x+24# .k -,=_4
,0! same r1son+1 if ;x is sli<tly small]
?an #2, !n ! frac;n is r\<ly
  ?5_/4(-tiny)(-1)#
    .k -?5_/4#*?1_/-tiny#
                                     #cc
    .k -?5_/4#(-,huge) .k +,huge,
s
  "lim%x $o #2^+]  ?x^3"-7x+11_/4x^2"
    -20x+24# .k +,=_4

,-b9+ te*niques4
  ,"o f9al example3"s (! cases we h
look$ at c appe> 9 ! same pro#m4 ,=
9/.e1 suppose we want$ 6-pute
  "lim%x $o #1] ?x^2"+2x-3_/x^2"-2x+1#_4
,we wd />t \ 0optimi/ic,y plu7+ 9
x .k #1, & wd f9d t we got ?0_/0# z a
result4 ,we "!=e look = -mon factors1
"k+ t x-1 is ! prime suspect4 ,factor+ \
an x-1 f ! num]ator & denom9ator turns !
frac;n 9to
  ?x^2"+2x-3_/x^2"-2x+1#
    .k ?(x-1)(x+3)_/(x-1)^2"#
    .k ?x+3_/x-1#_4
,we "!=e "k t
  "lim%x $o #1] ?x^2"+2x-3_/x^2"-2x+1#
    .k "lim%x $o #1] ?x+3_/x-1#,
& we set ab -put+ ? second limit4

                                     #cd
  ,ag we plug 9 x .k #1, & ? "t we get
?5_/0#, : tells u ! func;n has a v]tical
asymptote at x .k #1_4 ,s n[ we imag9e
plu7+ 9 a numb] ;x j sli<tly bi7] ?an
#1_4 ,we wd get
  ?x+3_/x-1# ".k<*] ?4_/+tiny#
    .k +,huge,
s t
  "lim%x $o #1^+]  ?x^2"+2x-3_/x^2"-2x
    +1# .k "lim%x $o #1^+]  ?x+3_/x-1#
    .k +,=_4
,simil>ly1 if we imag9e plu7+ 9 a numb]
;x j sli<tly small] ?an #1, we f9d
  ?x+3_/x-1# ".k<*] ?4_/-tiny#
    .k -,huge,
s t
  "lim%x $o #1^-]  ?x^2"+2x-3_/x^2"-2x
    +1# .k "lim%x $o #1^-]  ?x+3_/x-1#
    .k -,=_4





                                     #ce