Transcript
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!"##$%& ()*( *+, *-& .$+(/+"$"% 0"+.(/$+% %".) ()*( 2 = 6 *+,lim→ 3 ( ) = 3 6.
1/+, (2)2!"#$%&"'(
3&.*"%&
*+,
*-& .$+(/+"$"% 0"+.(/$+% 4& 5+$4 ()*(6
72 8)& 9/:/( lim→ () &;/%(% *+, &? ()& 9/:/( 9*4%6
= l i m→ 3 ( ) l i m→ () = l i m→ 3 ⋅ l i m→ lim→ ⋅ lim→ () lim→ 3 ( )
@3? ()& 9/:/( 9*4 0$- %":%2A
@3? ()& 9/:/( 9*4 0$- #-$,".(%2A
@3&.*"%& *+, .$+(/+"$"%2A=3(2)(2)(2)
*+",#-. /0(
2
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=3(2)(2)(2) !"##$%& ()*( *+, *-& .$+(/+"$"% 0"+.(/$+% %".) ()*( 2 = 6 *+,
lim→ 3 ( ) = 3 6.
1/+, (2)2
B& C"%( ,&,".&,6
lim→ 3 ( ) = 3 2 2 ( 2 ) !">%(/("(/+D ()& E*9"&% D/E&+ *>$E& 4& D&(6
3 6 = 3 2 6 2 .
8)"%6 2 = 42 □
*+",#-. /0(
!"#$%&"' 12"'%&'$-34(
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F%& ()& ,&0/+/(/$+ $0 .$+(/+"/(? *+, %(*+,*-, #-$#&-(/&% $0 9/:/(% ($ %)$4 ()*(
= 2 3 2 /% .$+(/+"$"% *( &E&-? #$/+( $0 /(% ,$:*/+2!"#$%&"'(
8$ >&D/+G $>%&-E& ()*( ()& ,$:*/+ $0 /% ℝ\ 2 2B& +&&, ($ %)$4 ()*( 0$- &E&-? #$/+( ∈ ℝ \ 2 G ()&+ lim→ = ()2H+ ,&(*/96
72 8)& 9/:/( $0 ()& +":&-*($-
lim→ 2 3 &;/%(% *+, &&.*"%& ≠ 22 !$ 4& .$+.9",&G >? ()& 9/:/( 9*4 0$-
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L;#9*/+ 4)? ()& 0"+.(/$+
= 1 1 if ≠ 1,2 if = 1, /% ,/%.$+(/+"$"% *( ()& #$/+( = 1 . !"#$%&"'(
M$(& ()*( ()& 7J%/,&, 9/:/(% *-&
/+0/+/(&6
• lim→ () = ∞. •
lim→ () = ∞.
*+",#-. /6(
8)"% ()& 9/:/( lim→ () ,$&% +$(&;/%(2
B& %*? ()*( )*% !" $"%$"$&'($)*+"&$",$&- *(
= 12
N&-& /% ()& D-*#)6
()
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L;#9*/+ 4)? ()& 0"+.(/$+
= 1
if < 11/ if ≥ 1
/% ,/%.$+(/+"$"% *( ()& #$/+( = 12!"#$%&"'(
8)& ,&0/+/(/$+ $0 .$+(/+"$"% )*% I #*-(%2 O"- (*%5 )&-& /% ($ 4$-5 $"( 4)/.)
#*-( D$&% 4-$+D 0/-%(2
8$ >&D/+G +$(& ()*( = 1 /% ,&0/+/(&9? /+ ()& ,$:*/+ $0 , so that isn’t the#-$>9&:2
8)& +&;(
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L;#9*/+ 4)? ()& 0"+.(/$+
= 1
if < 11/ if ≥ 1
/% ,/%.$+(/+"$"% *( ()& #$/+( = 12!"#$%&"' 12"'%&'$-34(
8)/% 0"+.(/$+ /% ,&0/+&, >? (4$ ,/00&-&+( -"9&% $+ ()& (4$ %/,&% $0 = 1G %$ ($,&./,& 4)&()&- ()& 9/:/( lim→ () &;/%(% ()& :$%( +*("-*9 ()/+D 4/99 >& ($,&(&-:/+& ()& 7J%/,&, 9/:/(%2
*+",#-. /7(
.$/$& %0+/ &1' 2'%&3
8$ ()& 9&0( $0
= 1 ()& 0"+.(/$+ /% ,&(&-:/+&, >? ()& -"9&
1 G %$ /+
()/% .*%&6 lim→ () = lim→ 1 = 02.$/$& %0+/ &1' 0$41&3
8$ ()& -/D)( $0
= 1 ()& 0"+.(/$+ /% ,&(&-:/+&, >? ()& -"9&
1/G %$ /+ ()/%
.*%&6 lim→ () = lim→ 1 / = 12
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L;#9*/+ 4)? ()& 0"+.(/$+
= 1
if < 11/ if ≥ 1
/% ,/%.$+(/+"$"% *( ()& #$/+( = 12!"#$%&"' 12"'%&'$-34(
B& C"%( ,&(&-:/+&, ()& 7J%/,&,9/:/(%2 B& 0$"+,6
• lim→ () = 02• lim→ = 1.
*+",#-. /7(
!$ &E&+ ()$"D) ()& 7J%/,&, 9/:/(% &;/%(G
()&? *-& ,/00&-&+(2 8)"% ()& 9/:/(lim→ () doesn’t exist.
N&-& /% ()& #9$(6 ()
□H+ $()&- 4$-,%G ()/% 0"+.(/$+ )*% * 5,/6
($)*+"&$",$&- *(
= 12
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*+",#-. /8(
L;#9*/+ 4)? ()& 0"+.(/$+
= 1 if ≠ 11 if = 1 /% ,/%.$+(/+"$"% *( ()& #$/+( = 12!"#$%&"'(
We’ll understand this function more clearly if we factorize and divide first:
= (1)(1)(1) i f ≠ 1
1 if = 1
= 1 i f ≠ 11 if = 1 .
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*+",#-. /8(
L;#9*/+ 4)? ()& 0"+.(/$+
= 1 if ≠ 11 if = 1 /% ,/%.$+(/+"$"% *( ()& #$/+( = 12!"#$%&"' 12"'%&'$-34(
M$4 4& .*+ ,&(&-:/+& ()& 9/:/(6
lim→ () = lim→ 1 @3&.*"%& ()&%& (4$ 0"+.(/$+% *D-&&*( &E&-? #$/+( &;.( *( 1G ()& #$/+(4)&-& 4& *-& (*5/+D ()& 9/:/(2A
=1/2. !$ ()& 9/:/( &;/%(% *% → 1G >"( /( /% ,/00&-&+( ($ ()& E*9"& ()*( ()& 0"+.(/$+(*5&% ()&-&G 4)/.) /%
12 H+ $()&- 4$-,%G ()& 0"+.(/$+ )*% * 0'/+7!82'
($)*+"&$",$&- *(
= 12 □
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*+",#-. /9(
B)*( E*9"& ,$&% )*E& ($ >& %$ ()*( ()& 0$99$4/+D 0"+.(/$+ /% .$+(/+"$"% *(&E&-? #$/+(P
= 2 5 3 3 if ≠ 3 if = 3 !"#$%&"'(
We’ll understand this function more clearly if we factorize the top line first:
= 2 5 3 3 i f ≠ 3
if = 3
= (21)(3) 3 if ≠ 3 if = 3
2 1 if ≠ 3
if 3
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*+",#-. /9(
B)*( E*9"& ,$&% )*E& ($ >& %$ ()*( ()& 0$99$4/+D 0"+.(/$+ /% .$+(/+"$"% *(&E&-? #$/+(P
= 2 5 3 3 if ≠ 3 if = 3 !"#$%&"' 12"'%&'$-34(
B& C"%( -&4-$(& ()/% 0"+.(/$+ /+ ()& 0$99$4/+D 4*?6
= 2 1 if ≠ 3 if = 3 B& %)$"9, %#9/( ()&
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*+",#-. /9(
B)*( E*9"& ,$&% )*E& ($ >& %$ ()*( ()& 0$99$4/+D 0"+.(/$+ /% .$+(/+"$"% *(&E&-? #$/+(P
= 2 5 3 3 if ≠ 3 if = 3 !"#$%&"' 12"'%&'$-34(
9!)' ;3 = 32H+ ()/% .*%&G ()& &
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*+",#-. /:(
L;#9*/+ 4)? %$:& %(*+,*-, ()&$-&:% *>$"( .$+(/+"$"% 0"+.(/$+% /:#9? ()*(
()& 0$99$4/+D 0"+.(/$+ /% .$+(/+"$"% *( &E&-? #$/+( $0 /(% ,$:*/+6
= tan4 . !"#$%&"'(
We’ll begin by working out exactly what the domain is. 8)& ,$:*/+ 4/99 >& ()& %&( $0 #$/+(% %*(/%0?/+D ()& I .$+,/(/$+% ()*(6
"# 9/&% /+ ()& ,$:*/+ $0 tanG 4)/.) /% ℝ \ … , , , , , … 2$#
9/&% /+ ()& ,$:*/+ $0
4 G 4)/.) /%
[2,2]2
I2 8)& ,&+$:/+*($- /% +$( K&-$ *( ()*(
2
B& ,&,".& ()*( dom = (2,2)\ , 2
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!"#$%&"' %" /:; 2"'%&'$-3(
Q&( >& *+ *->/(-*-? #$/+( 0-$: d o m = 2 , 2 \ , 21$- ()/% 4& ,&,".&672 8)& 0"+.(/$+ 4 /% * #$9?+$:/*9G %$G >? * %(*+,*-,
()&$-&:G /( /% .$+(/+"$"% *( &E&-? #$/+( $0 ℝ2 H+ #*-(/."9*-G /( /%.$+(/+"$"% *( 2
=2 B& 5+$4 ()*( ()& % 0G /%.$+(/+"$"% *( 4 2 8)"%G >? ()& %(*+,*-, ()&$-&: *>$"(()& .$:#$%/(/$+ $0 (4$ .$+(/+"$"% 0"+.(/$+% >&/+D
.$+(/+"$"%G 4& ,&,".& ()*( ()& .$:#$%/(/$+ 4 /%
.$+(/+"$"% *( ()& #$/+(
2
I2 1/+*99?G >&.*"%& () /% ()&
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*+",#-. />(
F%& .$+(/+"/(? ($ &E*9"*(& ()& 9/:/(6
lim→ s i n s i n .
!"#$%&"'(
F%/+D %(*+,*-, #-$#&-(/&% 4& .$:#"(&6
lim→ s i n s i n
= sin lim→ s i n @3&.*"%& sin /% .$+(/+"$"%G *+,?$" *-& *99$4&, ($ #*%% * 9/:/(/+($ * .$+(/+"$"% 0"+.(/$+2A
= s i n s i n @3&.*"%&
s i n /% ()& %": $0
(4$ .$+(/+"$"% 0"+.(/$+%G %$ /( /%.$+(/+"$"%G *+, %$ ()& 9/:/( *( &;/%(% *+, /% &;*.(9? &
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*+",#-. /?(
1/+, E*9"&% $0 *+, ()*( :*5& ()& 0$99$4/+D 0"+.(/$+ .$+(/+"$"% *( &E&-?#$/+(2
= 4 2 if < 2 3 if 2 ≤ < 32 if ≥ 3!"#$%&"'(
The only points where the function isn’t obviously continuous are the points = 2 *+, = 32 @% = A8)& 7J%/,&, 9/:/(% *(
= 2 *-&6
•
lim→ () = lim→ −− = l i m→( 2 ) = 42• lim→ () = lim→( 3 ) = 4 2 328)& 0"+.(/$+ )*% * 9/:/( *( = 2G *+, /% .$+(/+"$"% *( ()*( #$/+(G #-&./%&9? 4)&+()&%& (4$ &;#-&%%/$+% *-& &
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*+",#-. /?(
1/+, E*9"&% $0 *+, ()*( :*5& ()& 0$99$4/+D 0"+.(/$+ .$+(/+"$"% *( &E&-?#$/+(2
= 4 2 if < 2 3 if 2 ≤ < 32 if ≥ 3!"#$%&"' 12"'%&'$-34(
@% = A8)& 7J%/,&, 9/:/(% *( = 3 *-&6• lim→ () = lim→ 3 = 9 3 32•
lim→ () = lim→( 2 ) = 6 2
8)& 0"+.(/$+ )*% * 9/:/( *( = 3G *+, /% .$+(/+"$"% *( ()*( #$/+(G &;*.(9? 4)&+()&%& (4$ &;#-&%%/$+% *-& &
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*+",#-. /0B(
R$+%/,&- ()& 0"+.(/$+
=
1 0 s i n .
S-$E& ()*( ()&-& &;/%(% * -&*9 +":>&- %".) ()*( = 10002!"#$%&"'(
We’ll apply the I.V.T. to this function on the interval
[0,100]2
M$(& ()*(6
• 8)& 0"+.(/$+ /% .$+(/+"$"% *( &E&-? ∈ ℝ >&.*"%& /( /% * .$:>/+*(/$+$0 0"+.(/$+% @ *+, sinA ()*( *-& 5+$4+ ($ >& .$+(/+"$"%&E&-?4)&-&2
•
0 = 0
• 100 = 10000 10 sin 100 ≥ 99902• 0 = 0 < 1000 < 9990 = (100)2!$ ()& *%%":#(/$+% $0 ()& H2T282 *-& %*(/%0/&, $+ [0,100]G 4/() = 1 0 0 02B& .$+.9",& ()*( ()&-& &;/%(% * ∈[0 100] %".) ()*( 10002 □
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*+",#-. /00(
U&,".& "%/+D ()& H2T282 ()*( ()&-& &;/%(% * +":>&- ∈(0,1) 4)&-& = 1 . !"#$%&"'(R$+%/,&- ()& 0"+.(/$+ = 12We’ll apply the I.V.T. to this function on the interval [0,1] 4/() = 02
□
8)& 0/-%( ()/+D 4& )*E& ($ .)&.5 /% ()*(
() /% .$+(/+"$"% $+
[0,1]2
8)/% /% (-"& >&.*"%&• lim→ () = (0) • lim→ () = () 0$- *99 ∈(0,1]2
@V&.*99 ()*( 4& "%& * 7J%/,&, 9/:/( *(
= 0 >&.*"%&
() /% $+9? ,&0/+&, $+
()& -/D)( %/,& $0 02AM&;( 4& .*9."9*(&6• 0 = 12• 1 = 1.
!$ >&.*"%&
0 < 0 < (1) 4& ,&,".& 0-$: ()& H2T282 ()*( ()&-& &;/%(% *
4)&-& 02 8)/% 4/99 %$9E& ()& D/E&+ &
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*+",#-. /05(
H% ()& 0$99$4/+D 0"+.(/$+ .$+(/+"$"% *( *+? #$/+(P
= 0 if is rational, 1 if is irrational .
!"#$%&"'(
We’ll prove that there is no ∈ ℝ *( 4)/.) ()& 9/:/( lim→ () &;/%(%2 8)"% ()&0"+.(/$+ /% +$( .$+(/+"$"% *( *+? #$/+(2
First we’ll think of an intuitive explanation of this fact. After that we’ll give a
0$-:*9 #-$$02
3$() &;#9*+*(/$+% 4/99 ,&+, $+ ()& 0$99$4/+D >*%/. 0*.( *>$"( ℝ6CD&-2
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*+",#-. /05(
H% ()& 0$99$4/+D 0"+.(/$+ .$+(/+"$"% *( *+? #$/+(P
= 0 if is rational, 1 if is irrational .
!"#$%&"'(
First, an intuitive explanation of why the limit doesn’t exist:
H0 ()& 9/:/( ,/, /+ 0*.( &;/%( *( %$:& #$/+(
G ()&+6
• 8)&-& 4$"9, &;/%( * -&*9 +":>&- … • *+, ()&-& 4$"9, &;/%( * #$%/(/E& -&*9 … • %".) ()*( 0$- &E&-? #$/+( %".) ()*( 0 < < … • /( 4$"9, >& (-"& ()*( <
2
B)? /% ()*( /:#$%%/>9& 0$- ()/% ()P
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!"#$%&"' 12"'%&'$-34(
N&-& /% * #/.("-& $0 ()& %/("*(/$+6
= 1
= B)&+
∉ ℚG
()
9/&% $+ ()/% 9/+&2
B)&+ ∈ ℚG () 9/&% $+ ()/% 9/+&2
= 0 = /
= /
According to the definition of “ lim
→ () = ”, there exists a neighbourhood
*-$"+, G ( , ) \ G %".) ()*( ()& D-*#) $E&- ()$%& #$/+(% 9/&% &+(/-&9?>&(4&&+ ()& D-&&+ 9/+&%2But the “basic fact” we met earlier says that this set will definitely contain:
• *( 9&*%( $+& -*(/$+*9 +":>&- G• *+, *( 9&*%( $+& /--*(/$+*9 +":>&-
2
(, 0) (, 1)
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!"#$%&"' 12"'%&'$-34(
N&-& /% * D&+&-*9 #/.("-& $0 ()& %/("*(/$+6
= 1
= B)&+
∉ ℚG
()
9/&% $+ ()/% 9/+&2
B)&+ ∈ ℚG () 9/&% $+ ()/% 9/+&2
= 0 = /
= /
B)? /% ()/% /:#$%%/>9&PP
(, 0) (, 1)
8)/% /% /:#$%%/>9& >&.*"%&6
• 8)& 4/,() $0 ()& /+(&-E*9 0-$: / ($ / /% ½2• 3"( ()/% /+(&-E*9 /% %"##$%&, ($ .$E&- "#$% ()& 9/+& = 0 *% 4&99 *% ()& 9/+& = 1 @>&.*"%& (, 0) *+, (, 0) *-& >$() %"##$%&, ($ 9/& /+ /(2A• 8)/% /% /:#$%%/>9& )$4&E&- – ()& /+(&-E*9 /% $+9?
½ 4/,&G 4)/.) /% +$( 4/,&
&+$"D) ($ .$E&- 8+&1 $0 ()&%& 9/+&%2
!$ ()& 9/:/( .*++$( &;/%(2
12 1
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!"#$%&"' 12"'%&'$-34(
Now let’s convert this intuitive picture into a formal proof.
@0++% &1!& &1'0' $) "+
?1'0' &1' 2$/$&
lim→ () 'A$)&)B
W%%":&G $+ ()& $()&- )*+,G ()*( *+, *-& -&*9 +":>&-% 4)&-& lim→ = . H( 0$99$4%G *..$-,/+D ($ ()& ,&0/+/(/$+ $0 9/:/(G ()*( ()&-& &;/%(% * > 0 %".) ()*(&E&-? %*(/%0?/+D 0 < < )*% ()& #-$#&-(? ()*( < (⋆)2C!*&3 1$- *+? (4$ +":>&-% *+, 0-$: ()/% %&( ∈ ℝ | 0 < < G ()&+ < 12 ⋆⋆ . 8)/% 0$99$4% 0-$: ()& (-/*+D9& /+&
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@0++% &1!& &1'0' $) "+ ?1'0' &1' 2$/$& lim→ () 'A$)&) D*+"&$",'(EBB& C"%( ,&,".&, ()*( /0 *+, *-& -&*9 +":>&-% 4)&-& lim→ = ()&+ ()&-&&;/%(% *
> 0 %".) ()*( 0$- *+?
*+,
%*(/%0?/+D
0 < < ()&+
() < 12 (⋆⋆). M$4 4& .*+ ,&-/E& * .$+(-*,/.(/$+6
72 W..$-,/+D ($ ()& >*%/. 0*.( *>$"(
ℝ ()*( 4& %(*(&, &*-9/&-G 4& 5+$4 ()*(
()&-& 4/99 ,&0/+/(&9? &;/%( * &'$(#)'* *+, *+ (&&'$(#)'* /+ ()/% %&( ∈ ℝ | 0 < < 2=2 1$- ()&%& #$/+(%G = 0 *+, = 12 8)"%6 = 1. I2 8)/% /% * .$+(-*,/.(/$+ 4/()
(⋆⋆)2 □
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*+",#-. /06(
H% ()& 0$99$4/+D 0"+.(/$+ .$+(/+"$"% *( *+? #$/+(P
= 0 if is rational, if is irrational.
!"#$%&"'(
8)/% 0"+.(/$+ /% .$+(/+"$"% *( &;*.(9? $+& #$/+(6 = 028$ .$:#9&(&9? C"%(/0? ()/%G 4& )*E& ($ &;#9*/+6
72 B)? /% .$+(/+"$"% *( = 02=2 B)? /% +$( .$+(/+"$"% *( *+? ≠ 02F1- $) () *+"&$",+,) !& = 0 GB& .*+ &*%/9? *##9? ()& %&.*"%& 0 = 0G 4& D&( lim
→ () = 0 = (0)2
H+ $()&- 4$-,%G ()& 0"+.(/$+ ( ) /% .$+(/+"$"% *( 02
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!"#$%&"' %" *+",#-. /06 12"'%&'$-34(
O+& %$9"(/$+ 4$"9, >& ($ *,*#( ()& #-$$0 $0 ()& #-&E/$"% #-$>9&: ($ ()/%0"+.(/$+2 8)/% /% #$%%/>9& >"( /% *.("*99? 9&:G *+, ()&+ *##9? ()& 0'),2& $0 ()*( #-$>9&:2 @M$(&
()*( ()/% /% * ,/00&-&+( *##-$*.) ($ C"%( *,*#(/+D ()& #-$$02A
To do this we’ll need different names for the two functions. Denote the
$+& 4& *-& ."--&+(9? %(",?/+D >? = 0 if is rational, if is irrational.
Why isn’t () *+"&$",+,) !& !"- ≠ 0 G
W+, ,&+$(& ()& 0"+.(/$+ 4& %(",/&, /+ ()& #-&E/$"% #-$>9&: >? = 0 if is rational, 1 if is irrational H,')&$+"3 B)*( /% * 0$-:"9* -&9*(/+D ()&%& (4$ 0"+.(/$+%P
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!"#$%&"' %" *+",#-. /06 12"'%&'$-34(
I")?'03
H,')&$+"3 B)*( /% * 0$-:"9* -&9*(/+D ()&%& (4$ 0"+.(/$+%P
1$- &E&-? ≠ 0G = 2H,')&$+"3 B)? ,$&% ()/% /:#9? ()*(
() /% +$( .$+(/+"$"% *( *+?
≠ 0 P
I")?'03
W%%":&G $+ ()& $()&- )*+,G ()*( () 4*% *.("*99? .$+(/+"$"% *( %$:& ≠ 02 B&99G ()& 0"+.(/$+ 4$"9, >& .$+(/+"$"% *( *% 4&99G 4/() * +$+JK&-$ E*9"& ()&-&2
H( 4$"9, ()&+ 0$99$4 0-$: * %(*+,*-, ()&$-&: *>$"( .$+(/+"$"%
0"+.(/$+% ()*( ()&
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*+",#-. /07(
U$&% ()&-& &;/%( * +":>&- 4)/.) 7 D-&*(&- ()*+ /(% .">&P
!"#$%&"'(
H+ $()&- 4$-,%6 /% ()&-& * ∈ ℝ %".) ()*( = 1 PH( /% &+$"D) ($ #-$E& ()*( ()&-& &;/%(% * ∈ ℝ %".) ()*( ()& 0"+.(/$+
=
1 (*5&% ()& E*9"&
() = 02
To deduce this we’ll apply the I.V.T. to () $+ ()& /+(&-E*9 [2,0]2M$(& ()*(6
• () /% * #$9?+$:/*9G %$ /( /% .$+(/+"$"% *( &E&-? #$/+(2•
2 = 52
• 0 = 1. • 2 < 0 < (0)2!$ 4& .*+ ,&,".& 0-$: ()& H2T282 ()*( %".) * 4/99 /+,&&, &;/%(2
□
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*+",#-. /08(
!"#$%&"'(
S-$E& ()*( 0$- *+? #*/- $0 #$%/(/E& +":>&-% *+, ()& 0$99$4/+D&&9$4G +$(& ()*( 0 622
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!"#$%&"' 12"'%&'$-34(
We’ll focus on the following piece of the domain of ()6 , 1
N$4 ,/, 4& ,&,".& ()&%& = 9/:/(%P R$+%/,&- ()& 0/-%( $+&G 0$- &;*:#9&2
O>%&-E& ()*( *% → +…
O>%&-E& ()& 0$99$4/+D 9/:/(%6
• lim→ −+ () = ∞2• lim→ = ∞2
= 1 1 ∗ 12 54 1 1 ∗ 12 74
8)/% #/&.& *##-$*.)&%
%$:& 0/+/(& #$%/(/E&
+":>&-2
8)/% #/&.&
*##-$*.)&%
+#,($(-. ()/()($01
8)/% #/&.& *##-$*.)&%
%$:& 0/+/(& +&D*(/E&
+":>&-2
8)/% #/&.& *##-$*.)&%
%$:& 0/+/(& #$%/(/E&
+":>&-2
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!"#$%&"' 12"'%&'$-34(
We’ll focus on the following piece of the domain of ()6 , 1
H( 0$99$4% 0-$: ()&%& 9/:/(% ()*( ()&-& &;/%( +":>&-% *+, %".) ()*( 12 52 < < < 1 %".) ()*(
• > 0, • < 02
O>%&-E& ()& 0$99$4/+D 9/:/(%6
• lim→ −+ () = ∞2•
lim→ = ∞2
B& 0/+/%) >? *##9?/+D ()& H2T282 ($ ()& 0"+.(/$+ () $+ ()& /+(&-E*9 [,] 4/()
02 □
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*+",#-. /08 1#$
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*+",#-. /09(
!"#$%&"' %" 1D4(
*2 !)$4 ()*( ()& *>%$9"(& E*9"& 0"+.(/$+ /% .$+(/+"$"% *( &E&-?#$/+(2
>2 L;#9*/+ 4)? /0 %$:& 0"+.(/$+
() /% .$+(/+"$"% $+ %$:& /+(&-E*9G
()&+ %$ /% | |2.2 H0 | | /% .$+(/+"$"%G ,$&% /( 0$99$4 ()*( /% .$+(/+"$"%P
8)& *>%$9"(& E*9"& 0"+.(/$+ .*+ >& 4-/((&+6 = if ≤0, if ≥0. 8)& $+9? #$/+( 4& -&*99? +&&, ($ .)&.5 .*-&0"99? /% = 02W( ()/% #$/+(6
• lim→ = l i m→()=0. • lim→ | | = l i m→ = 0 . 3&.*"%& ()& 7J%/,&, 9/:/(% >$() &;/%(G *+, *-& &
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!"#$%&"' %" 1,4(
W%%":& ()*( () /% .$+(/+"$"% $+ %$:& /+(&-E*9 28)&+
| | /% .$+(/+"$"% $+ ()& %*:& /+(&-E*9
>&.*"%& /( /% #-&./%&9?
()& *+/6+)$&$+" $0 ()& *>%$9"(& E*9"& 0"+.(/$+ || 4/() ()G *+, 4&5+$4 ()*( ()&-& /% * %(*+,*-, ()&$-&: ()*( ()& .$:#$%/(/$+ $0 (4$.$+(/+"$"% 0"+.(/$+% /% .$+(/+"$"%2
!"#$%&"' %" 124(
The converse doesn’t hold. In other words, continuous doesn’t imply()*( /% .$+(/+"$"%2W %/:#9& .$"+(&-&;*:#9& /%6
= 1 if ≥ 01 i f "( | | /% ()&.$+%(*+( 0"+.(/$+ 7G 4)/.) .9&*-9? /% .$+(/+"$"% *( &E&-? #$/+(2 □
* ,# /0:
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*+",#-. /0:(
*+""= (
R$+%/,&- 0"+.(/$+% *+, G *+, -&*9 +":>&-% *+, 2W%%":& ()*(6
"# () /% .$+(/+"$"% *( 2=2 8)& 9/:/( lim→ ()=. S-$E& ()*(6 lim
→ ( ) = lim
→() .
Let’s start by writing out explicitly the information we were given in the
*%%":#(/$+% $0 ()& ()&$-&:6
72 8)&-& /% * -"9&
%".) ()*( 0$- &E&-?
> 0G 4)&+&E&-
%*(/%0/&%
0 < < ()G ()&+ < 2=2 8)&-& /% * -"9& %".) ()*( 0$- &E&-? > 0G 4)&+&E&- %*(/%0/&%0 < < ()G ()&+ < 2
= 1 & 34
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*+""= 12"'%&'$-34 (
7A 8)&-& /% * -"9&
%".) ()*( 0$- &E&-?
> 0G 4)&+&E&-
%*(/%0/&%
< ()G ()&+ < 2=A 8)&-& /% * -"9& %".) ()*( 0$- &E&-? > 0G 4)&+&E&- %*(/%0/&%0 < < ()G ()&+ < 2
F1!& ?' J"+?3
F1!& ?' !0' &0-$"4 &+ (+3
F%& ()/% /+0$-:*(/$+ ($ 0/+, * -"9& () %".) ()*( 0$- &E&-? > 0G4)&+&E&- %*(/%0/&% 0 < < G ()&+ < 2
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72 8)&-& /% * -"9& %".) ()*( 0$- &E&-? > 0G 4)&+&E&- %*(/%0/&% < ()G ()&+ < 2=2 8)&-& /% * -"9&
%".) ()*( 0$- &E&-?
> 0G 4)&+&E&-
%*(/%0/&%
0 < < ()G ()&+ < 2
F1!& ?' J"+?3
K1' 60++%
Q&(
>& *+ *->/(-*-? #$%/(/E& -&*92
!&( = ( ) 3 . Q&( >& *+ *->/(-*-? -&*9 %*(/%0?/+D 0 < < ( 4 ) . 8)&+G %">%(/("(/+D (3) *+, (4) /+($ (2) 4& ,&,".&6
< 5 .
8)&+ 4& ,&,".& 0-$: (1) @-*./+D >? ()A ()*(6 ( ) () < .□
* ,# /0>
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*+",#-. /0>(
*+""= (
O"- (*%5 /% ($ #-$E& ()*( (4$ %(*(&:&+(% *-& &
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*+""= 12"'%&'$-34(
⇒ ?
• =&!&'/'"& IB H0 * 0"+.(/$+ () /% .$+(/+"$"% *( * #$/+( G *+, * 0"+.(/$+ () )*% * 9/:/(&&.*"%& $0 @7AA
*+, ∈ dom() @>&.*"%& $0 @=AA2B)? /% lim→ ∘ () = ( ) Plim→ ( ) = lim→ ()
( )
( ) @8)& *%%":#(/$+% $0 W *-& %*(/%0/&, )&-&2A
@3? @7A2A
*+""= 12"'%&'$-34
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*+""= 12"'%&'$-34(
⇒ ?
• =&!&'/'"& IB H0 * 0"+.(/$+ () /% .$+(/+"$"% *( * #$/+( G *+, * 0"+.(/$+ () )*% * 9/:/(&
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*+""= 12"'%&'$-34(
⇒ ? c o n t i n u e d .
• =&!&'/'"& IB H0 * 0"+.(/$+ () /% .$+(/+"$"% *( * #$/+( G *+, * 0"+.(/$+ () )*% * 9/:/(&
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