You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
òîáòî α(õ) ³ β(õ) íà íåñê³í÷åííîñò³ º íåñê³í÷åííî ìàë³ ôóíêö³¿.<br />
Çíàéäåìî<br />
α( x)<br />
β( x)<br />
x<br />
x<br />
−2<br />
−1<br />
lim = lim =<br />
1 lim x =<br />
−<br />
x→∞ x→∞ x→∞<br />
0 .<br />
Îòæå, α(õ) º íåñê³í÷åííî ìàëîþ ôóíêö³ºþ âèùîãî ïîðÿäêó<br />
ìàëèçíè, í³æ β(õ), êîëè õ ïðÿìóº äî íåñê³í÷åííîñò³.<br />
Çì³ñòîâí³ ïðèêëàäè íåñê³í÷åííî ìàëèõ ôóíêö³é îäíàêîâîãî<br />
ïîðÿäêó ìàëèçíè áóäå ðîçãëÿíóòî òðîõè ï³çí³øå.<br />
6.2.3. Äâ³ âàæëèâ³ ãðàíèö³<br />
6.2.3 à. Ïåðøà âàæëèâà ãðàíèöÿ. Äîâåäåìî, ùî<br />
sin x<br />
lim = 1 . (6.2.3)<br />
x → 0 x<br />
Ä î â å ä å í í ÿ áóäåìî ïðîâîäèòè ïîåòàïíî.<br />
1 0 . Äîâåäåííÿ ïîäâ³éíî¿ íåð³âíîñò³<br />
p<br />
sin x< x< tg x, 0 < x< . (6.2.4)<br />
2<br />
2 0 . Äîâåäåííÿ íåð³âíîñò³<br />
sin x £ x ,<br />
p<br />
x < .<br />
2<br />
(6.2.5)<br />
3 0 . Äîâåäåííÿ ð³âíîñò³<br />
4 0 . Äîâåäåííÿ ð³âíîñò³<br />
limsin x = 0 . (6.2.6)<br />
x →0<br />
lim cos x = 1 . (6.2.7)<br />
x →0<br />
5 0 . Çàâåðøåííÿ äîêàçó ð³âíîñò³ (6.2.3).<br />
1 0 . Ðîçãëÿíåìî äóãó êîëà ðàä³óñà R = 1 ç öåíòðàëüíèì<br />
êóòîì, ðàäiàííà ì³ðà ÿêîãî äîð³âíþº x æ ö<br />
p<br />
ç 0< < çè<br />
x 2÷<br />
ø<br />
(ðèñ. 6.6).<br />
Òîä³ 0À = 1, sin x = MK, tg x = AT.<br />
Ðèñ. 6.6<br />
Î÷åâèäíî, ùî ïëîùà òðèêóòíèêà 0ÀÌ ìåíøà ïëîù³ ñåêòîðà<br />
0ÀÌ, êîòðà â ñâîþ ÷åðãó ìåíøà ïëîù³ òðèêóòíèêà<br />
0ÀÒ. Çàâäÿêè öüîìó ôàêòó îòðèìàºìî òàêó ïîäâ³éíó íåð³âí³ñòü<br />
1 1 1<br />
sin x< x< tg x .<br />
2 2 2<br />
²ç îñòàííüî¿ íåð³âíîñò³ ³ âèïëèâຠíåð³âí³ñòü (6.2.4).<br />
2 0 . Ó òî÷ö³ õ = 0 ìຠì³ñöå ð³âí³ñòü (0 = 0). ßêùî õ çàäîâîëüíÿº<br />
íåð³âí³ñòü 0 < x < , òî çã³äíî ç 1 0 ìàºìî<br />
p<br />
2<br />
sin x < x àáî sin x < x .<br />
Òåïåð ðîçãëÿíåìî âèïàäîê, êîëè x æ p ö<br />
Î- ç ,0<br />
çè 2 ø÷<br />
.<br />
Òîä³ î÷åâèäíî, ùî<br />
sin x =-sin( - x) = sin( - x)
Change language