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Çàóâàæåííÿ 1. Ïåðøå îçíà÷åííÿ ãðàíèö³ ôóíêö³¿<br />
íàçèâàºòüñÿ îçíà÷åííÿì çà Ãåéíå 1 . ²íêîëè êàæóòü, ùî ïåðøå<br />
îçíà÷åííÿ ãðàíèö³ ôóíêö³¿ äàºòüñÿ íà ìîâ³ ïîñë³äîâíîñò³.<br />
Çàóâàæåííÿ 2. Äðóãå îçíà÷åííÿ ãðàíèö³ ôóíêö³¿ íàçèâàºòüñÿ<br />
îçíà÷åííÿì çà Êîø³ 2 . ²íêîëè êàæóòü, ùî äðóãå<br />
îçíà÷åííÿ ãðàíèö³ ôóíêö³¿ äàºòüñÿ íà ìîâ³ ”ε−δ”.<br />
Òåîðåìà 6.2.1. Îçíà÷åííÿ ãðàíèö³ ôóíêö³¿ çà Ãåéíå òà<br />
çà Êîø³ åêâ³âàëåíòí³, òîáòî ç ïåðøîãî îçíà÷åííÿ âèïëèâàº<br />
äðóãå ³ íàâïàêè.<br />
Ïðè äîñë³äæåíí³ ôóíêö³é äóæå âàæëèâèì º ïîíÿòòÿ<br />
îäíîñòîðîíí³õ ãðàíèöü.<br />
Îçíà÷åííÿ 6.2.3. ×èñëî À 1 (À 2 ) íàçèâàºòüñÿ ãðàíèöåþ<br />
ôóíêö³¿ y = f(x) ñïðàâà (çë³âà) ïðè õ → õ 0 , x > õ 0 (x < õ 0 ),<br />
ÿêùî ôóíêö³ÿ âèçíà÷åíà â ïðàâîìó (ë³âîìó) δ-îêîë³ òî÷êè<br />
õ 0 ³ äëÿ ∀ε > 0 ∃ δ(ε) > 0, ùî ∀ õ òàêèõ, ùî<br />
õ 0 < x < õ 0 + δ (õ 0 – δ < x < õ 0 ) âèêîíóºòüñÿ íåð³âí³ñòü<br />
|f(x) – A 1 |0∃δ> 0: òàêå, ùî ∀ õ 0 M. Ïîçíà÷àºòüñÿ lim fx ( ) =∞.<br />
x→x0<br />
Òåîðåìè, ðîçãëÿíóò³ â ï. 5.2.2 ïðî ãðàíèö³ ïîñë³äîâíîñòåé,<br />
ñïðàâåäëèâ³ ³ äëÿ ôóíêö³é. Çîêðåìà, ñïðàâåäëèâ³ òàê³<br />
òåîðåìè.<br />
Òåîðåìà 6.2.2. ßêùî ôóíêö³ÿ ìຠãðàíèöþ, òî âîíà<br />
ºäèíà.<br />
Òåîðåìà 6.2.3. Íåõàé ³ñíóþòü<br />
Òîä³ ñïðàâåäëèâ³ òàê³ òâåðäæåííÿ:<br />
lim fx ( ) = A ³<br />
x→x0<br />
1)–2) lim ( fx ( ) ±ϕ ( x) ) = A± B;<br />
3) lim ( )<br />
x→x0<br />
cf x = c f x = cA c −<br />
x→x0<br />
4) lim ( ( )) lim ( ) , ñòàëà;<br />
5)<br />
x→x0 x→x0<br />
fx ()<br />
= A , B ≠0.<br />
x x B<br />
lim<br />
→ 0 ϕ () x<br />
lim ϕ ( x)<br />
= B .<br />
x→x0<br />
fx ( ) ⋅ϕ ( x) = A⋅B;<br />
Òåîðåìà 6.2.4. Íåõàé ³ñíóþòü ãðàíèö³ lim fx ( ),<br />
lim ψ ( x)<br />
,<br />
x→x0<br />
x→x0<br />
ÿê³ äîð³âíþþòü îäíîìó ³ òîìó ñàìîìó ÷èñëó A. Êð³ì öüîãî,<br />
â îêîë³ òî÷êè õ = x 0 , çà âèíÿòêîì, ìîæëèâî, ñàìî¿ òî÷êè, ìàº<br />
ì³ñöå ïîäâ³éíà íåð³âí³ñòü<br />
Òîä³ ³ñíóº ãðàíèöÿ<br />
x→x0<br />
fx ( ) ≤ϕ( x) ≤ψ ( x)<br />
. (6.2.2)<br />
lim ϕ ( x)<br />
, ³ âîíà òåæ äîð³âíþº ÷èñëó A,<br />
òîáòî lim ϕ ( x)<br />
= A.<br />
x→x0<br />
Äîâåäåìî îäíó ç öèõ òåîðåì, íàïðèêëàä 6.2.4.<br />
Äîâåäåííÿ áóäåìî ïðîâîäèòè êîðèñòóþ÷èñü îçíà÷åííÿì<br />
ãðàíèö³ ôóíêö³¿ çà Ãåéíå. Ðîçãëÿíåìî äîâ³ëüíó<br />
ïîñë³äîâí³ñòü {x n }, ÿêà ïðÿìóº äî x 0 (x n ≠ x 0 ). Ïðè öüîìó<br />
çàâäÿêè ïîäâ³éí³é íåð³âíîñò³ (6.2.2) âîíà íàðîäæóº åëåìåíòè<br />
ïîñë³äîâíîñòåé çíà÷åíü ôóíêö³é fx (<br />
n), j( xn), y ( xn)<br />
, ÿê³<br />
çàäîâîëüíÿþòü òàêó ïîäâ³éíó íåð³âí³ñòü<br />
fx ( ) £j ( x) £y( x ).<br />
n n n<br />
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