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Çã³äíî ç îçíà÷åííÿì ãðàíèö³ ôóíêö³¿ çà Ãåéíå ³ ó â³äïîâ³äíîñò³<br />
äî óìîâ òåîðåìè åëåìåíòè fx (<br />
n), y ( xn)<br />
â³äïîâ³äíèõ<br />
ïîñë³äîâíîñòåé ïðÿìóþòü äî ÷èñëà À. Òåïåð çàñòîñóºìî òåîðåìó<br />
5.2.5 ïðî òðè ïîñë³äîâíîñò³. Íà îñíîâ³ ¿¿ îòðèìàºìî,<br />
ùî ³ñíóº ãðàíèöÿ lim ϕ ( x)<br />
³, á³ëüø òîãî, lim ϕ ( x)<br />
= A. Òåîðåìó<br />
x→x0<br />
x→x0<br />
äîâåäåíî.<br />
Òâåðäæåííÿ òåîðåìè 6.2.3 ïðîïîíóºìî äîâåñòè ÷èòà÷åâ³<br />
ñàìîñò³éíî.<br />
Ìîæíà òàêîæ ðîçãëÿäàòè ãðàíèö³ ôóíêö³é ïðè x → ±∞<br />
(ãðàíèö³ ôóíêö³é íà íåñê³í÷åííîñò³).<br />
Ïðèïóñòèìî, ùî ôóíêö³ÿ y = f(x) âèçíà÷åíà, íàïðèêëàä,<br />
íà ìíîæèí³ âñ³õ äîäàòíèõ ä³éñíèõ ÷èñåë, ³ íåõàé àðãóìåíò<br />
õ íåîáìåæåíî çðîñòຠ(öå çàïèñóþòü òàê: x → +∞). Ïðè öüîìó<br />
ìîæå ñòàòèñÿ òàê, ùî ïðè íåîáìåæåíîìó çðîñòàíí³ õ<br />
çíà÷åííÿ ôóíêö³¿ ìîæå íàáëèæàòèñÿ äî äåÿêîãî ÷èñëà À.<br />
Ó öüîìó âèïàäêó ÷èñëî À íàçèâàþòü ãðàíèöåþ ôóíêö³¿ ïðè<br />
x → +∞.<br />
Îçíà÷åííÿ 6.2.5. ×èñëî À íàçèâàþòü ãðàíèöåþ ôóíêö³¿<br />
ïðè x → +∞, ÿêùî äëÿ áóäü-ÿêîãî äîäàòíîãî ÷èñëà ε ³ñíóº<br />
òàêå äîäàòíå ÷èñëî ∆ + , ùî ç íåð³âíîñò³ x > ∆ + âèïëèâຠíåð³âí³ñòü<br />
|f(x) –A| < ε.<br />
Òîé ôàêò, ùî ÷èñëî À º ãðàíèöåþ ôóíêö³¿ f(x) ïðè<br />
x → +∞, çàïèñóþòü òàê:<br />
limfx ( ) = A.<br />
x →∞<br />
Àíàëîã³÷íî îçíà÷àºòüñÿ ãðàíèöÿ ôóíêö³¿ ïðè x →−∞.<br />
Îçíà÷åííÿ 6.2.6. ×èñëî  íàçèâàþòü ãðàíèöåþ ôóíêö³¿<br />
ïðè x →−∞, ÿêùî äëÿ áóäü-ÿêîãî äîäàòíîãî ÷èñëà ε ³ñíóº<br />
òàêå â³ä’ºìíå ÷èñëî ∆ − , ùî äëÿ âñ³õ õ < ∆ − âèïëèâຠíåð³âí³ñòü<br />
|f(x) –B| < ε.<br />
Òîé ôàêò, ùî ÷èñëî B º ãðàíèöåþ ôóíêö³¿ f(x) ïðè<br />
x →−∞, çàïèñóþòü òàê:<br />
lim fx ( ) = B.<br />
x→−∞<br />
6.2.2. Ïîð³âíÿííÿ íåñê³í÷åííî ìàëèõ âåëè÷èí<br />
Ïðè äîñë³äæåíí³ ïîâåä³íêè íåñê³í÷åííî ìàëî¿ ôóíêö³¿ ó<br />
äàí³é òî÷ö³ â ìàòåìàòè÷íîìó àíàë³ç³ ³ñíóº äîñèòü åôåêòèâíèé<br />
ìåòîä, ÿêèé áàçóºòüñÿ íà ïîð³âíÿíí³ äîñë³äæóâàíî¿<br />
íåñê³í÷åííî ìàëî¿ ôóíêö³¿ ç â³äîìîþ (åòàëîííîþ) íåñê³í-<br />
÷åííî ìàëîþ ôóíêö³ºþ ó âèãëÿä³ ãðàíèö³ â³äíîøåííÿ ¿õ. Ó<br />
çàëåæíîñò³ â³ä òîãî, ÷îìó äîð³âíþº öÿ ãðàíèöÿ, íåñê³í÷åííî<br />
ìàëèì ôóíêö³ÿì äàþòü ïåâíó íàçâó.<br />
Ïîäàìî òàê³ ïîçíà÷åííÿ. Íåõàé α(õ) ³ β(õ) º íåñê³í÷åííî<br />
ìàë³ ôóíêö³¿ â òî÷ö³ õ 0 ∈(à, b); ÿê ³ ðàí³øå, õ 0 ìîæå áóòè é<br />
íåâëàñòèâîþ (x 0 = ∞).<br />
Îçíà÷åííÿ 6.2.7. Ôóíêö³ÿ α(õ) íàçèâàºòüñÿ íåñê³í÷åííî<br />
ìàëîþ ôóíêö³ºþ âèùîãî ïîðÿäêó ìàëèçíè, í³æ ôóíêö³ÿ<br />
α( x)<br />
β(õ), ÿêùî lim = 0 . Ïðè öüîìó β(õ) íàçèâàºòüñÿ íåñê³íx<br />
→ x 0 β ( x )<br />
÷åííî ìàëîþ íèæ÷îãî ïîðÿäêó ìàëèçíè, í³æ α(õ). Öåé ôàêò<br />
ñèìâîë³÷íî ïîçíà÷àºòüñÿ òàê: α(x) =o(β(x)), êîëè x → x 0 (÷èòàºòüñÿ<br />
“î-ìàëå”).<br />
Îçíà÷åííÿ 6.2.8. Ôóíêö³¿ α(õ) ³ β(õ) íàçèâàþòüñÿ íåñê³í-<br />
÷åííî ìàëèìè îäíàêîâîãî ïîðÿäêó ìàëèçíè, ÿêùî<br />
α<br />
lim ( x )<br />
x → x ( )<br />
0 β x<br />
= c , äå ñ º â³äì³ííå â³ä íóëÿ ÷èñëî. Öåé ôàêò ñèìâîë³÷íî<br />
ïîçíà÷àºòüñÿ òàê: α(x) =Î(β(x)), êîëè x → x 0 (÷èòà-<br />
ºòüñÿ “Î-âåëèêå”). ßêùî æ ñ = 1, òî α(õ) ³ β(õ) íàçèâàþòüñÿ<br />
â òî÷ö³ õ 0 åêâ³âàëåíòíèìè, ³ çàïèñóþòü α(õ) ∼ β(õ).<br />
Ïðèêëàä 6.2.6. Íåõàé α(õ) =õ 2 , β(õ) =õ. Òîä³ α(õ) ³ β(õ)<br />
â òî÷ö³ õ = 0 º íåñê³í÷åííî ìàë³ ôóíêö³¿. Çíàéäåìî<br />
2<br />
α( x)<br />
x<br />
lim = lim = lim x = 0 .<br />
x → 0 β( x)<br />
x → 0 x x → 0<br />
Îòæå, α(õ) º íåñê³í÷åííî ìàëà ôóíêö³ÿ âèùîãî ïîðÿäêó<br />
ìàëèçíè, í³æ β(õ) ïðè x → 0.<br />
Ïðèêëàä 6.2.7. Íåõàé α(õ) =õ −2 , β(õ) =õ −1 , òîä³<br />
lim α ( x) = 0, limβ ( x) = 0,<br />
x→∞<br />
x→∞<br />
176 177
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