Γιώργος Μαυρίδης-Όρια και συνέχεια

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Επιλεγμένες, ωραίες ασκήσεις από το ανανεωμένο βιβλίο του Γιώργου Μαυρίδη για την Γ΄ Λυκείου, το οποίο δίνει μεγάλη βαρύτητα στις πολλές λυμένες ασκήσεις που περιέχει.

Transcript of Γιώργος Μαυρίδης-Όρια και συνέχεια

Page 1: Γιώργος Μαυρίδης-Όρια και συνέχεια

ISBN: 978-618-80214-8-8

Page 2: Γιώργος Μαυρίδης-Όρια και συνέχεια

nȍȨȦ ȰȪȵȦȨ�ȩȨȮ�ȚȯȢȞȪȷȦȞȭȞ�ȭȷȪȚ� ȜȢȚ�ȦȚ ȣȚȭȚȦȨȗȬȞȭȞ�ȭȢȫ�țȚȬȢȣȖȫ�ȖȦȦȨȢȞȫ� ȣȚȢ�ȦȚ�ȞȥțȚȡȶȦȞȭȞ�ȬȭȨȦ�ȭȪȵȩȨ�ȬȣȖȱȠȫ� ȣȚȢ ȞȪȜȚȬȘȚȫ��ȡȚ�ȭȨȦ�ȣȞȪȝȘȬȞȭȞ� ȩȨȤȤȚȩȤȕȬȢȨ�ȚȪȜȵȭȞȪȚ.»

ȁȞȵȝȲȪȨȫ�Ȇ��ȃȚȟȚȦȭȟȗȫ

Page 3: Γιώργος Μαυρίδης-Όρια και συνέχεια

ƲƳƱƭƱƥƱƴ

ĬĮ� ȝʌȠȡȠȪıĮȞ� IJĮ� ȂĮșȘȝĮIJȚțȐ� ȞĮ� ĮʌȠIJİȜȑıȠȣȞ� ȝȑȡȠȢ� IJȠȣȜȐȤȚıIJȠȞ� IJȘȢ�ĮʌȐȞIJȘıȘȢ�ıIJȠȞ�ȚįİȫįȘ�țȩıȝȠ�IJȠȣ��ȥȑȝĮIJȠȢ�ȩʌȠȣ�IJĮ�ʌȐȞIJĮ�įȘȜȫȞȠȣȞ�țĮȚ�įȘȜȫ�ȞȠȞIJĮȚ�ȦȢ�ĮȜȘșȚȞȐ��

ĬĮ�ȝʌȠȡȠȪıĮȞ� IJĮ�ȂĮșȘȝĮIJȚțȐ�ȞĮ�ĮʌȠIJİȜȑıȠȣȞ� IJȘȞ�țĮȪıȚȝȘ�ȪȜȘ�Ș�ȠʌȠȓĮ�șĮ�İȞİȡȖȠʌȠȚȠȪıİ�IJȠȞ�ȞĮȡțȦȝȑȞȠ�ȖȓȖĮȞIJĮ�IJȘȢ�ıȣȞİȓįȘıȘȢ��

ĬĮ� ȝʌȠȡȠȪıĮȞ� IJĮ� ȂĮșȘȝĮIJȚțȐ�� ȦȢ� ȖȜȫııĮ� IJȘȢ� ȜȠȖȚțȒȢ�� ȞĮ� ıȣȞįȡȐȝȠȣȞ�ĮʌȠijĮıȚıIJȚțȐ�IJȚȢ�ıIJȡȚȝȦȖȝȑȞİȢ�įȣȞȐȝİȚȢ�IJȘȢ�ıȪȞİıȘȢ��IJȘȢ�IJĮʌİȚȞȩIJȘIJĮȢ��IJȘȢ�ȖȞȫ�ıȘȢ�țĮȚ�IJȘȢ�ĮȜȒșİȚĮȢ��Įʌȩ�IJĮ�ȝĮȞȚĮıȝȑȞĮ�ıIJȓijȘ�IJȠȣ�©șȑȜȦª��IJȠȣ�©ijĮȓȞȠȝĮȚª��IJȠȣ�©ȑȤȦª��IJȠȣ�©ȖȡȒȖȠȡĮª��IJȠȣ�©țĮIJĮȞĮȜȫȞȦª��IJȠȣ�©İȞIJȣʌȦıȚĮțȠȪª��IJȠȣ�©İȖȫª�

ȅ�țĮșȑȞĮȢ�Įʌ¶�ȩıȠȣȢ�ȕȡȓıțȠȣȞ�ȗȦȠȖȩȞȠ�IJȠ�ȞĮ�ĮıȤȠȜȠȪȞIJĮȚ�ıİ�ȕȐșȠȢ�ȝİ�IJȘ�ȗȦȒ��șĮ�ȑįȚȞİ�IJȘ�įȚțȒ�IJȠȣ�ĮʌȐȞIJȘıȘ�

ǻȪıțȠȜĮ�ʌȐȞIJȦȢ�șĮ�ȣʌȒȡȤĮȞ�įȚĮijȦȞȓİȢ��ıIJȠ�ȩIJȚ�șĮ�ȒIJĮȞ��ĮįȪȞĮIJȠȞ��ĮțȩȝȘ�țĮȚ�ȖȚĮ�IJĮ�ȂĮșȘȝĮIJȚțȐ�ȞĮ�İʌȚIJİȜȑıȠȣȞ�ȑȡȖȠ�ıİ�țȐʌȠȚȠ�Įʌȩ�IJĮ�ʌȡȠȘȖȠȪȝİȞĮ��ĮȞ�țȐIJȚ�IJĮ�ĮʌȑțȠʌIJİ�Įʌȩ�IJȘȞ�ȑȞȞȠȚĮ�IJȡȠijȩ�IJȠȣȢ��Įʌȩ�IJȘȞ�ȑȞȞȠȚĮ�IJȠȣ�ĮʌİȓȡȠȣ�

ǻİȞ�ȞȠȝȓȗȦ�ȩIJȚ�ȤȡİȚȐȗİIJĮȚ�Ș�ıȣȞįȡȠȝȒ�IJȘȢ�țȕĮȞIJȠȝȘȤĮȞȚțȒȢ�� IJȘȢ�İȟİȜȚțIJȚ�țȒȢ�ȕȚȠȜȠȖȓĮȢ�Ȓ�IJȘȢ�șİȦȡȓĮȢ�IJȠȣ�ȤȐȠȣȢ��ȫıIJİ�ȞĮ ȠȡȚıIJİȓ�ȦȢ ȜȠȖȚțȒ�ĮȞĮțȠȜȠȣșȓĮ���Ș�ıȪȞįİıȘ�IJȠȣ�ĮʌİȓȡȠȣ�IJȦȞ�įȣȞĮIJȫȞ�ʌİȡȚʌIJȫıİȦȞ��ȝİ�IJȘȞ�IJȣȤȠįȚȦțIJȚțȠȪ�IJȪʌȠȣ��İʌȚȜȠȖȒ�țȐʌȠȚȦȞ�Įʌȩ�ĮȣIJȑȢ�

Ȃ¶�ĮȣIJȒ� ĮțȡȚȕȫȢ� IJȘȞ� ʌİʌȠȓșȘıȘ� ĮʌȑijȣȖĮ� İʌȚȝİȜȫȢ�� țȐșİ� ȠȜȠțȜȘȡȦIJȚțȠȪ�IJȪʌȠȣ�ȠȝĮįȠʌȠȓȘıȘ�IJȦȞ�ĮıțȒıİȦȞ��ǹȣIJȩ�įȘȜĮįȒ�ʌȠȣ�IJȩıȠ�İȪıIJȠȤĮ�Ƞ�ĮȟȑȤĮıIJȠȢ�įȐıțĮȜȠȢ�ĬİȩįȦȡȠȢ�ȀĮȗĮȞIJȗȒȢ�ȤĮȡĮțIJȒȡȚȗİ�ȦȢ�©ʌİȡȚʌIJȦıȚȠȜȠȖȓĮª�

ǻİȞ�ȑȤİȚ�ȞȩȘȝĮ�ȞĮ�ĮȞĮȗȘIJȒıİȚ�țĮȞİȓȢ�IJȚȢ�ĮȚIJȓİȢ�ĮȣIJȒȢ�IJȘȢ�İȝȝȠȞȒȢ��İȐȞ�įȘ�ȜĮįȒ�ȠijİȓȜİIJĮȚ�ıİ� ȑȜȜİȚȝȝĮ�ȜȠȖȚțȒȢ�Ȓ� İȞIJȚȝȩIJȘIJĮȢ��ȆȐȞIJȦȢ�� ȝȠȣ� İȓȞĮȚ� ȚįȚĮȓIJİȡĮ�įȪıțȠȜȠ�ȞĮ�ĮʌȠijȪȖȦ�IJȠȞ�ʌİȚȡĮıȝȩ�țȐʌȠȚȦȞ�İȞȠȤȜȘIJȚțȫȞ�ĮȜȜȐ�ȝȐȜȜȠȞ�ʌȡĮȖȝĮIJȚ�țȫȞ� ıțȑȥİȦȞ��ȦȢ� ʌȡȠȢ� IJĮ� ĮʌȠIJİȜȑıȝĮIJĮ� ĮȣIJȫȞ� IJȦȞ� ĮȞIJȚȜȒȥİȦȞ� Ȓ� ȓıȦȢ� “ȠȡijĮ�ȞȫȞ” ʌȡĮțIJȚțȫȞ�

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ȀȐșİ�ȥİȣįĮȓıșȘıȘ�țĮȚ�ȝȚĮ�ijȣȜĮțȒ���ȀȐʌȦȢ�ȑIJıȚ�țȚ�İįȫ�Ș�ȥİȣįĮȓıșȘıȘ�IJȘȢ�ıȣıIJȘȝĮIJȚțȒȢ�ȠȡȖȐȞȦıȘȢ�țĮȚ�ȝİȜȑIJȘȢ�ijȣȜĮțȓȗİȚ�IJȠ�įȡȠıİȡȩ�ʌȞİȪȝĮ�IJȦȞ�ȝĮșȘIJȫȞ��ıIJİȡȫȞIJĮȢ�Įʌ¶�ĮȣIJȠȪȢ�IJȘ�ȝĮșȘIJİȓĮ�IJȘȢ�İȟİȡİȪȞȘıȘȢ��IJȘȞ�ĮȞİȝİȜȚȐ�IJȘȢ�ʌİȡȚʌȜȐȞȘ�ıȘȢ�țĮȚ�IJȘ�ȤĮȡȐ�IJȘȢ�ĮȞĮțȐȜȣȥȘȢ��ȈIJİȡȫȞIJĮȢ�ijȣıȚțȐ�ıIJȘ�ıȣȞȑȤİȚĮ�ıIJȠ�ȕĮșȝȩ�ʌȠȣ�IJĮ�İʌȘȡİȐȗȠȣȞ��IJĮ�ȝĮșȘȝĮIJȚțȐ�ĮȜȜȐ�țȣȡȓȦȢ�IJȘȞ�țȠȚȞȦȞȓĮ��Įʌȩ�ȝȣĮȜȐ�İȟİȡİȣȞȘȝȑ�ȞĮ�țĮȚ�ȐȡĮ�İȟİȡİȣȞȘIJȚțȐ��ȝİ� IJİȜȚțȩ�ĮʌȠIJȑȜİıȝĮ�IJȘȞ�ȤȦȡȓȢ�ĮȞIJȓʌĮȜȠ�İʌȚțȡȐIJȘıȘ�IJȘȢ�ȝȓȝȘıȘȢ��IJȘȢ�ijĮȚįȡȩIJȘIJĮȢ�țĮȚ�IJȘȢ�İʌĮȞȐȜȘȥȘȢ��ȅȚ�ĮʌȠįİȓȟİȚȢ�ȩȜȦȞ�IJȦȞ�ʌȡȠȘ�ȖȠȪȝİȞȦȞ��ʌȡȠijĮȞȫȢ�įİȞ�ȤȡİȚȐȗİIJĮȚ�ȞĮ�ĮȞĮȗȘIJȘșȠȪȞ ıİ�ıțȠȞȚıȝȑȞĮ�ȝĮșȘȝĮIJȚțȐ�ȕȚȕȜȓĮ��ǹȞ�IJİȜȚțȐ�ijȠȕȓȗİȚ�IJȩıȠ�ʌȠȜȪ�ȝȚĮ�ȝĮIJȚȐ�ȝȑıĮ�ȝĮȢ��IJȠȣȜȐȤȚıIJȠȞ�IJȫȡĮ�İȓȞĮȚ�ĮȡțİIJȒ�ȝȚĮ�ȝĮIJȚȐ�įȓʌȜĮ�ȝĮȢ�

ĬİııĮȜȠȞȓțȘ��ǿȠȪȞȚȠȢ 2013 īȚȫȡȖȠȢ ȁ��ȂĮȣȡȓįȘȢ

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ƲƧƳƫƧƸƱƮƧƯƣ ƣƘ�ƵƱƮƱƶ

ȆȡȩȜȠȖȠȢ�

ȂȚȖĮįȚțȠȓ�ĮȡȚșȝȠȓ ........................................................................................ 11 � ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 13 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 113 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 141 ǵȡȚȠ�–�ȈȣȞȑȤİȚĮ�ıȣȞȐȡIJȘıȘȢ ....................................................................... 151� ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 153 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 347 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 391 �ǻȚĮijȠȡȚțȩȢ�ȁȠȖȚıȝȩȢ��ǹǯ�ȂȑȡȠȢ� ............................................................... 401 ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 403 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 477 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 501 ǹʌĮȞIJȒıİȚȢ .................................................................................................... 507 �ǺȚȕȜȚȠȖȡĮijȓĮ ................................................................................................ 569

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ƲƧƳƫƧƸƱƮƧƯƣ ƤƘ�ƵƱƮƱƶ

ȆȡȩȜȠȖȠȢ� ǻȚĮijȠȡȚțȩȢ�ȁȠȖȚıȝȩȢ��Ǻǯ�ȂȑȡȠȢ� ............................................................... 11� ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 13 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 259 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 317 ȅȜȠțȜȘȡȦIJȚțȩȢ�ȁȠȖȚıȝȩȢ ........................................................................... 329 ȁȣȝȑȞĮ�șȑȝĮIJĮ ........................................................................................ 331 ȆȡȠIJİȚȞȩȝİȞĮ�șȑȝĮIJĮ .............................................................................. 505 ǻȚĮȖȦȞȓıȝĮIJĮ .......................................................................................... 543 ǼʌĮȞĮȜȘʌIJȚțȐ�ĬȑȝĮIJĮ ................................................................................ 555 ǹʌĮȞIJȒıİȚȢ .................................................................................................... 575 �ǺȚȕȜȚȠȖȡĮijȓĮ ................................................................................................ 636

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Page 8: Γιώργος Μαυρίδης-Όρια και συνέχεια

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ȂǼĬȅǻȅȁȅīǿǹ�īȞȦȡȓȗȠȣȝİ� ȩIJȚ� Ș� ıȣ�ȞȐȡIJȘıȘ f İȓȞĮȚ�ȖȞȘıȓȦȢ�ȝȠȞȩIJȠȞȘ��ȅʌȩIJİ��ȖȚĮ�ȞĮ�ȕȡȠȪȝİ� IJȠ� İȓįȠȢ� IJȘȢ�ȝȠ�ȞȠIJȠȞȓĮȢ� IJȘȢ�� Įȡțİȓ� ȞĮ�ıȣȖțȡȓȞȠȣȝİ� įȪȠ� IJȚȝȑȢ�IJȘȢ� � �1f x țĮȚ� � �2f x ȝİ� 1 2x x .�

Ȉȋȅȁǿȅ�ǹʌȠįİȓȟĮȝİ ȩIJȚ� Ș� ıȣ�ȞȐȡIJȘıȘ�f İȓȞĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ��ǹȞ� ȝʌȠȡȑıȠȣ�ȝİ� ȞĮ� ȕȡȠȪȝİ� țĮȚ� ıİ�ʌȠȚȠ�ıȘȝİȓȠ�ȝȘįİȞȓȗİIJĮȚ��IJȩIJİ� İȪțȠȜĮ� șĮ� ȝʌȠȡȑ�ıȠȣȝİ� ȞĮ� ȕȡȠȪȝİ� IJȠ�ʌȡȩıȘȝȩ�IJȘȢ�

Page 9: Γιώργος Μαυρίδης-Όρια και συνέχεια

ƟǒNJǐ-ƴǖǎƾǘdžNJǂ�ƴǖǎƽǒǕLjǔLjǓ� � 337�

iv)� H ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ʌİȡȚIJIJȒ��ǻȘȜĮįȒ� � � � �f x f x� � ȖȚĮ�țȐșİ�� x .�\

ǼʌȠȝȑȞȦȢ�

� � � �x xlim f x lim f xo�f o�f

� �ª º¬ ¼

� �xlim f x 4.o�f

� �

ǹȞ� șȑıȠȣȝİ� u x � � ʌĮȡĮIJȘȡȠȪȝİ� ȩIJȚ�

ȩIJĮȞ�IJȠ�x IJİȓȞİȚ�ıIJȠ� ��f IJȩIJİ�IJȠ�u IJİȓ�

ȞİȚ�ıIJȠ� .�f ȅʌȩIJİ��Ș�ȚıȩIJȘIJĮ

� �xlim f x 4o�f

� �

ȖȡȐijİIJĮȚ

� �ulim f u 4o�f

įȘȜĮįȒ�

� �xlim f x 4.o�f

Ǿ�ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ıȣȞİȤȒȢ�țĮȚ�ȖȞȘıȓȦȢ�ĮȪȟȠȣıĮ�ıIJȠ� .\ ǼʌȠȝȑȞȦȢ�

� � � � � �� �x xI OLP I [ � OLP I [

o�f o�f \ � ��� � � �

ȈǾȂǼǿȍȈǾ�Ǿ�ıȣȞȐȡIJȘıȘ� f İȓȞĮȚ�ıȣȞİȤȒȢ�țĮȚ�ȖȞȘıȓȦȢ� ĮȪȟȠȣıĮ� ıIJȠ įȚȐıIJȘȝĮ

� �� � �f � f\ ǼʌȠȝȑȞȦȢ�

� � � � � �� �x xI OLP I [ � OLP I [ �

o�f o�f \

DZȡĮ�� Ș� İȪȡİıȘ� IJȠȣ� ıȣȞȩȜȠȣ� IJȚ�ȝȫȞ� IJȘȢ� � �f \ ĮȞȐȖİIJĮȚ� ıIJȠȞ�

ȣʌȠȜȠȖȚıȝȩ�IJȠȣ�ȠȡȓȠȣ� � �xlim f xo�f

ĮijȠȪ� IJȠ� ȩȡȚȠ� � �xlim f xo�f

İȓȞĮȚ�

ȖȞȦıIJȩ�

Page 10: Γιώργος Μαυρίδης-Όρια και συνέχεια

338� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

ƪƾǍǂ�137�

ǻȓȞİIJĮȚ� IJȡȓȖȦȞȠ�ǹȅǺ�ȝİ� � � � �$ �� � � 2 �� � � țĮȚ� � � �% �� � � � � ȃĮ� ĮʌȠįİȓȟİIJİ�ȩIJȚ�ȣʌȐȡȤİȚ�ıȘȝİȓȠ�Ȃ�ıIJȠ�İıȦIJİȡȚțȩ�IJȠȣ�ȪȥȠȣȢ�IJȠȣ�Ǻǻ�IJȑIJȠȚȠ��ȫıIJİ�

� � � � � �02 0$ 0% � � ��ƭǞǔLj� Ǿ� İȣșİȓĮ� Ǻǻ� įȚȑȡȤİIJĮȚ� Įʌȩ� IJȠ� ıȘȝİȓȠ�� �% �� � țĮȚ� İȓȞĮȚ� țȐșİIJȘ� ıIJȘȞ� İȣșİȓĮ� ȅǹ�

įȘȜĮįȒ��ıIJȠȞ�ȐȟȠȞĮ� y y.c DZȡĮ��ȑȤİȚ�İȟȓıȦıȘ y 2. � ȅʌȩIJİ�� Įȡțİȓ ȞĮ� ĮʌȠįİȓȟȠȣȝİ� ȩIJȚ� ȣʌȐȡȤİȚ� � ��[ �� �� IJȑIJȠȚȠȢ��ȫıIJİ�IJȠ�ıȘȝİȓȠ�

� ��0 [ � � ȞĮ�İʌĮȜȘșİȪİȚ�IJȘ�ıȤȑıȘ

� � � � � �MO MA MB �

įȘȜĮįȒ

� � � �222 2 2 2� � �[ � [ � [ � �� � � � � �

Ȓ�ȚıȠįȪȞĮȝĮ

2 2� � �[ � [ � [ � ��� � � � �

ȆȡȠȢ�IJȠȪIJȠ��șİȦȡȠȪȝİ�IJȘ�ıȣȞȐȡIJȘıȘ

� � 2 2I [ [ � [ � [ � � � � � � � > @[ �� ��

țĮȚ�ʌĮȡĮIJȘȡȠȪȝİ�ȩIJȚ� � Ǿ��f İȓȞĮȚ�ıȣȞİȤȒȢ�ıIJȠ�įȚȐıIJȘȝĮ� > @�� � ȦȢ�ĮʌȠIJȑȜİıȝĮ�ʌȡȐȟİȦȞ�ȝİIJĮȟȪ�ıȣȞİȤȫȞ�

ıȣȞĮȡIJȒıİȦȞ � � �I � � � � � � � � � � � țĮȚ�� � �I � �� �� � �� �� � � � � ! .

ȅʌȩIJİ���ıȪȝijȦȞĮ�ȝİ�IJȠ�șİȫȡȘȝĮ�IJȠȣ�Bolzano� ȣʌȐȡȤİȚ� � ��[ �� �� IJȑIJȠȚȠȢ��ȫıIJİ�

� ��I [ � � 2 2� � �[ � [ � [ � ��� � � � �

Ȉȋȅȁǿȅ�Ǿ�įȣıțȠȜȓĮ�IJȠȣ�ʌȡȠȕȜȒȝĮIJȠȢ�İıIJȚȐ�ȗİIJĮȚ� ıIJȘȞ� ĮțȡȚȕȒ� ȝİIJȐijȡĮıȒ� IJȠȣ�Įʌȩ� IJȘ� ȖȜȫııĮ� IJȘȢ� īİȦȝİIJȡȓĮȢ� ıIJȘ�ȖȜȫııĮ�IJȘȢ�ǹȞȐȜȣıȘȢ�

y

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Page 11: Γιώργος Μαυρίδης-Όρια και συνέχεια

ƟǒNJǐ-ƴǖǎƾǘdžNJǂ�ƴǖǎƽǒǕLjǔLjǓ� � 339�

ƪƾǍǂ�138�

DzıIJȦ�ıȣȞȐȡIJȘıȘ� f : o\ \ � Ș� ȠʌȠȓĮ� İȓȞĮȚ� 1 1� � țĮȚ� ıȣȞİȤȒȢ��ǹȞ� ȖȚĮ� țȐ�ʌȠȚȠȞ�ʌȡĮȖȝĮIJȚțȩ�ĮȡȚșȝȩ�Ȝ�ȚıȤȪȠȣȞ�ȠȚ�ıȤȑıİȚȢ�

� � � �ȜI � I � �� �����țĮȚ����

��� � � � � � �2Ȝ � I � ȜI � ��� � �IJȩIJİ��i)� ȞĮ�ȕȡİȓIJİ�IJȘȞ�IJȚȝȒ�IJȠȣ�Ȝ�ii)� ȞĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�Ș�İȟȓıȦıȘ� � �f x 0 �ȑȤİȚ�ĮțȡȚȕȫȢ�ȝȓĮ�ȡȓȗĮ�Ș�ȠʌȠȓĮ�

ȝȐȜȚıIJĮ�įİȞ�İȓȞĮȚ�ĮțȑȡĮȚȠȢ�ĮȡȚșȝȩȢ����ƭǞǔLj� i)� ȆĮȡĮIJȘȡȠȪȝİ�ȩIJȚ�IJȠ�ȗİȪȖȠȢ

� � � � � �� �[� \ I � � I �

ĮʌȠIJİȜİȓ�ȜȪıȘ�IJȠȣ�ȖȡĮȝȝȚțȠȪ�ıȣıIJȒȝĮIJȠȢ

� �Ȝ[ \ �2Ȝ � [ Ȝ\ �� Ȝ �� ­

® � � �¯ \

ǵȝȦȢ�� IJȠ� ıȪıIJȘȝĮ� ĮȣIJȩ� ȑȤİȚ� İʌȓıȘȢ�� ȦȢ� ȜȪıȘ�țĮȚ�IJȠ�ȗİȪȖȠȢ

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ȀĮȚ�İʌİȚįȒ

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ĮijȠȪ�Ș�ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�1 1� ıȣȝʌİȡĮȓȞȠȣȝİ�ȩIJȚ� IJȠ� ʌĮȡĮʌȐȞȦ�ıȪıIJȘȝĮ� ȑȤİȚ� ȐʌİȚȡİȢ� ȜȪıİȚȢ��ǻȘȜĮįȒ�

Ȝ �

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Ȓ�ȚıȠįȪȞĮȝĮ

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ȈǾȂǼǿȍȈǾ�ǹȞ ȠȚ� ıIJĮșİȡȠȓ� ȩȡȠȚ�İȞȩȢ� ȖȡĮȝȝȚțȠȪ� ıȣıIJȒ�ȝĮIJȠȢ� 2 2u İȓȞĮȚ�țĮȚ�ȠȚ�įȪȠ� ȝȘįȑȞ�� IJȩIJİ� IJȠ� ıȪ�ıIJȘȝĮ�įİȞ� İȓȞĮȚ�ĮįȪȞĮ�IJȠȞ� ĮijȠȪ� ȑȤİȚ� ȝȓĮ� IJȠȣ�ȜȐȤȚıIJȠȞ� ʌȡȠijĮȞȒ� ȜȪ�ıȘ�IJȘȞ�

� � � �[� \ �� � �

ǼʌȠȝȑȞȦȢ�� șĮ� ȑȤİȚ� Ȓ�ȝȠȞĮįȚțȒ�ȜȪıȘ�IJȘȞ�

� � � �[� \ �� �

ȠʌȩIJİ� ' ��z

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Page 12: Γιώργος Μαυρίδης-Όρια και συνέχεια

340� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

ii)� ǹȞIJȚțĮșȚıIJȫȞIJĮȢ�IJȘȞ�IJȚȝȒ�IJȠȣ�Ȝ�ʌȠȣ ȕȡȒțĮȝİ�ʌĮȡĮʌȐȞȦ�ıİ�țȐʌȠȚĮ�Įʌȩ�IJȚȢ�įȠ�

șİȓıİȢ�ıȤȑıİȚȢ�ȑȤȠȣȝİ

� � � �I � I � �� � � � � �f 1 f 2 . �

ȈIJȘ�ıȣȞȑȤİȚĮ��ʌĮȡĮIJȘȡȠȪȝİ�ȩIJȚ�Ș�ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ıȣȞİȤȒȢ�ıIJȠ�įȚȐıIJȘȝĮ� > @�� �

țĮȚ�ȚıȤȪİȚ�

� � � � � � � � � �2I � I � I � I � I � � � � � .

ȆȡȐȖȝĮIJȚ�� ĮȞ� İȓȞĮȚ � �I � �� IJȩIJİ� șĮ� ȑȤȠȣȝİ� țĮȚ � �I � � �� ʌȡȐȖȝĮ� ĮįȪȞĮIJȠȞ�

ĮijȠȪ�Ș�ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�1 1.� DZȡĮ��ıȪȝijȦȞĮ�ȝİ�IJȠ�șİȫȡȘȝĮ�IJȠȣ�Bolzano Ș�İ�

ȟȓıȦıȘ � �I [ � ȑȤİȚ� ȝȓĮ� IJȠȣȜȐȤȚıIJȠȞ� ȡȓȗĮ� ıIJȠ� įȚȐıIJȘȝĮ� � ��� � � Ǿ� ȡȓȗĮ� ĮȣIJȒ�

ʌȡȠijĮȞȫȢ�įİȞ�İȓȞĮȚ�ĮțȑȡĮȚȠȢ�ĮȡȚșȝȩȢ�țĮȚ�ijȣıȚțȐ�İȓȞĮȚ�ȝȠȞĮįȚțȒ��ĮijȠȪ�Ș�ıȣȞȐȡ�

IJȘıȘ�f İȓȞĮȚ�1 1.�

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Page 13: Γιώργος Μαυρίδης-Όρια και συνέχεια

ƟǒNJǐ-ƴǖǎƾǘdžNJǂ�ƴǖǎƽǒǕLjǔLjǓ� � 341�

ƪƾǍǂ�139�

DzıIJȦ�ȠȚ�ȝȚȖĮįȚțȠȓ�ĮȡȚșȝȠȓ��z��țĮȚ���w��ȖȚĮ�IJȠȣȢ�ȠʌȠȓȠȣȢ�ȚıȤȪȠȣȞ�ȠȚ�ıȤȑıİȚȢ�2 2

2 z zz z z2�

� � �

țĮȚ��2w z z i. � � �

i)� ȃĮ� ȕȡİȓIJİ� IJȠȞ� ȖİȦȝİIJȡȚțȩ� IJȩʌȠ� IJȦȞ� İȚțȩȞȦȞ� IJȠȣ� z� ıIJȠ� ȝȚȖĮįȚțȩ���������İʌȓʌİįȠ��

ii)� ȃĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ�ȣʌȐȡȤİȚ�ȝȠȞĮįȚțȩȢ�ĮȡȚșȝȩȢ�z�IJȑIJȠȚȠȢ��ȫıIJİ�� �0 Im z 1� � ����țĮȚ��� � �Im w 0 .�

iii)� AȞ�Ƞ��w�İȓȞĮȚ�ʌȡĮȖȝĮIJȚțȩȢ��ȞĮ�ĮʌȠįİȓȟİIJİ�ȩIJȚ� z 2.� ���

ƭǞǔLj� i)� DzıIJȦ� z x yi � țĮȚ� � �0 [� \ Ș� İȚțȩȞĮ� IJȠȣ� ıIJȠ� ȝȚȖĮįȚțȩ� İʌȓʌİįȠ�� Ǿ� įȠșİȓıĮ�

ıȤȑıȘ

2 2

2 z zz z z2�

� �

ȚıȠįȪȞĮȝĮ�ȖȡȐijİIJĮȚ

� � � � � �2 222 2 x yi x yi

x y x yi x yi2

� � �� � � � �

2 2 2 2

2 2 x y 2xyi x y 2xyix y 2x2

� � � � �� � �

2 2

2 2 2x 2yx y 2x2�

� � � 2 2 2 2x y x y 2x� � � �

22y 2x� 2y x.�

DZȡĮ��Ƞ�ȗȘIJȠȪȝİȞȠȢ�ȖİȦȝİIJȡȚțȩȢ�IJȩʌȠȢ�İȓȞĮȚ�Ș�ʌĮȡĮȕȠȜȒ 2F � \ [�

Page 14: Γιώργος Μαυρίδης-Όρια και συνέχεια

342� � ƮǂljLjǍǂǕNJNjƽ�ƥƘ�ƭǖNjdžǀǐǖ�

ii)� DzȤȠȣȝİ � � � �22w z z i x yi x yi i � � � � � �

� � � �

2 2

2 2

x y 2xyi x yi i

x y x 2xy y 1 i.

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ǵȝȦȢ� ĮʌȠįİȓȟĮȝİ�ȩIJȚ 2y x . ȅʌȩIJİ�

� � � � � �� � � � � �w y y y 2y y y 1 i y 2y y 1 i ȝİ \ � � � � � � � � � �\

ǹȡțİȓ�ȜȠȚʌȩȞ�ȞĮ�ĮʌȠįİȓȟȠȣȝİ�ȩIJȚ�Ș�İȟȓıȦıȘ ��\ \ � �� �

ȑȤİȚ�ȝȠȞĮįȚțȒ�ʌȡĮȖȝĮIJȚțȒ�ȡȓȗĮ��Ș�ȠʌȠȓĮ�ĮȞȒțİȚ�ıIJȠ�įȚȐıIJȘȝĮ� � ���� � ȆȡȠȢ�IJȠȪ�

IJȠ��șİȦȡȠȪȝİ�IJȘ�ıȣȞȐȡIJȘıȘ � � �I \ �\ \ �� \ � � �\

țĮȚ� ʌĮȡĮIJȘȡȠȪȝİ� ȩIJȚ� İȓȞĮȚ� ȖȞȘıȓȦȢ� ĮȪȟȠȣıĮ�� ȆȡȐȖȝĮIJȚ�� ȖȚĮ� țȐșİ� 1 2\ � \ �\

ȝİ� 1 2y y� ȑȤȠȣȝİ � �1 22y 2y .� ȅʌȩIJİ��

� �1 1 2 22y y 1 2y y 1� � � � �

įȘȜĮįȒ�� � � � �1 2f y f y .�

ǼʌȓıȘȢ��Ș�ıȣȞȐȡIJȘıȘ�f İȓȞĮȚ�ıȣȞİȤȒȢ�ıIJȠ�\ ȐȡĮ�țĮȚ�ıIJȠ�įȚȐıIJȘȝĮ� > @��� ȝİ�

� � � �I � I � � �� � �

ǼʌȠȝȑȞȦȢ��ıȪȝijȦȞĮ�ȝİ�IJȠ�șİȫȡȘȝĮ�IJȠȣ�Bolzano Ș�İȟȓıȦıȘ�

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