problem set 5 solutions.pdf

8/19/2019 Problem Set 5 Solutions.pdf

1/42

!"##$%& ()*( *+, *-& .$+(/+"$"% 0"+.(/$+% %".) ()*( 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


2/42

=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(


3/42

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& .$+.9",&G >? ()& 9/:/( 9*4 0$-


4/42

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

()


5/42

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)& +&;(


6/42

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


7/42

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


8/42

*+",#-. /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 .


9/42

*+",#-. /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 □


10/42

*+",#-. /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


11/42

*+",#-. /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/( ()&


12/42

*+",#-. /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 ()& &


13/42

*+",#-. /:(

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


14/42

!"#$%&"' %" /:; 2"'%&'$-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 >&.*"%& () /% ()&


15/42

*+",#-. />(

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? &


16/42

*+",#-. /?(

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$ &;#-&%%/$+% *-& &


17/42

*+",#-. /?(

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$ &;#-&%%/$+% *-& &


18/42

*+",#-. /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 □


19/42

*+",#-. /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&+ &


20/42

*+",#-. /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


21/42

*+",#-. /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


22/42

!"#$%&"' 12"'%&'$-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)


23/42

!"#$%&"' 12"'%&'$-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


24/42

!"#$%&"' 12"'%&'$-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& /+&


25/42

@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 □


26/42

*+",#-. /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


27/42

!"#$%&"' %" *+",#-. /06 12"'%&'$-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


28/42

!"#$%&"' %" *+",#-. /06 12"'%&'$-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"+.(/$+% ()*( ()&


29/42

*+",#-. /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

□


30/42

*+",#-. /08(

!"#$%&"'(

S-$E& ()*( 0$- *+? #*/- $0 #$%/(/E& +":>&-% *+, ()& 0$99$4/+D&&9$4G +$(& ()*( 0 622


31/42

!"#$%&"' 12"'%&'$-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


32/42

!"#$%&"' 12"'%&'$-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 □


33/42

*+",#-. /08 1#$


34/42

*+",#-. /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 *+, *-& &


35/42

!"#$%&"' %" 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:


36/42

*+",#-. /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


37/42

*+""= 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


38/42

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>


39/42

*+",#-. /0>(

*+""= (

O"- (*%5 /% ($ #-$E& ()*( (4$ %(*(&:&+(% *-& &


40/42

*+""= 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


41/42

*+""= 12"'%&'$-34(

⇒ ?

• =&!&'/'"& IB H0 * 0"+.(/$+ () /% .$+(/+"$"% *( * #$/+( G *+, * 0"+.(/$+ () )*% * 9/:/(&


42/42

*+""= 12"'%&'$-34(

⇒ ? c o n t i n u e d .

• =&!&'/'"& IB H0 * 0"+.(/$+ () /% .$+(/+"$"% *( * #$/+( G *+, * 0"+.(/$+ () )*% * 9/:/(&

problem set 5 solutions.pdf

Documents