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Ñïðàâåäëèâ³ñòü ôîðìóë (6.3.8) — (6.3.14) âñòàíîâëþºòüñÿ<br />
çà äîïîìîãîþ âàæëèâèõ ãðàíèöü (6.3.6) — (6.3.7), òåîðåìè<br />
6.2.5 ³ ç âðàõóâàííÿì íåïåðåðâíîñò³ îñíîâíèõ åëåìåíòàðíèõ<br />
ôóíêö³é.<br />
Äîâåäåííÿ ñïðàâåäëèâîñò³ ôîðìóë (6.3.8) — (6.3.10),<br />
(6.3.12) — (6.3.14) ïîäàìî ó âèãëÿä³ ïðèêëàä³â.<br />
Ïðèêëàä 6.3.5. Âèâåñòè ôîðìóëó (6.3.8).<br />
Ðîçâ’ÿçàííÿ. Ïîêëàäåìî kx = t, òîä³ çã³äíî ç ôîðìóëîþ<br />
(6.3.6) îòðèìàºìî<br />
sinkx sin kx sint<br />
lim = klim = klim<br />
= k.<br />
x→0 x x→0 kx t→0<br />
t<br />
Ïðèêëàä 6.3.6. Âèâåñòè ôîðìóëó (6.3.9).<br />
Ðîçâ’ÿçàííÿ. Ïåðåéäåìî äî íîâî¿ çì³ííî¿ t = arcsin x,<br />
òîä³ ç âðàõóâàííÿì òîãî, ùî ïðè x® 0Þ t ® 0 , îòðèìàºìî<br />
arcsin x t<br />
lim = lim = 1 .<br />
x →0 x t →0<br />
sin t<br />
Ïðèêëàä 6.3.7. Âèâåñòè ôîðìóëó (6.3.12).<br />
Ðîçâ’ÿçàííÿ. Êîðèñòóþ÷èñü âëàñòèâ³ñòþ ëîãàðèôì³÷íî¿<br />
ôóíêö³¿, çàïèøåìî ôóíêö³þ, ÿêà çíàõîäèòüñÿ ï³ä çíàêîì<br />
ãðàíèö³ ó âèãëÿä³: log<br />
1<br />
a<br />
(1 + x) x . Âðàõîâóþ÷è öå çîáðàæåííÿ,<br />
íåïåðåðâí³ñòü ëîãàðèôì³÷íî¿ ôóíêö³¿ ³ ôîðìóëó<br />
(6.3.7), îòðèìàºìî<br />
+<br />
x<br />
log 1 1<br />
a<br />
(1 x) 1<br />
x x<br />
lim lim log<br />
a(1 x) log<br />
a lim(1 x) loga<br />
e<br />
x→0 x→0 x→0<br />
= + = + = = .<br />
ln a<br />
Ïðèêëàä 6.3.8. Âèâåñòè ôîðìóëó (6.3.13)<br />
Ðîçâ’ÿçàííÿ. Çàïèøåìî éîãî â òàê³é ôîðì³:<br />
x<br />
x<br />
a − 1 ⎡t = a − 1, x = log (1 + ) ⎤<br />
lim =<br />
a<br />
t<br />
t<br />
⎢<br />
⎥ = lim = ln a<br />
x→0 x → 0 ⇒ → 0<br />
t→0<br />
⎣x<br />
t<br />
⎦ log<br />
a (1 + t )<br />
.<br />
Ïðèêëàä 6.3.9. Âèâåñòè ôîðìóëó (6.3.14).<br />
Ðîçâ’ÿçàííÿ.<br />
µ<br />
µ<br />
(1 + x) − 1 ⎡t = (1 + x) −1, x→0 ⇒t<br />
→0⎤<br />
t<br />
lim = ⎢<br />
⎥ = lim =<br />
x→0 x ln(1 + ) =µ ln(1 + )<br />
x→0<br />
⎣ t x ⎦ x<br />
t ln(1 + x)<br />
=µ lim lim<br />
0<br />
0 ln(1 t )<br />
=µ<br />
x→<br />
t→<br />
+ x<br />
.<br />
×èòà÷åâ³ ðåêîìåíäóºìî ïðîêîìåíòóâàòè ðîçâ’ÿçàííÿ ïðèêëàä³â<br />
6.3.5 — 6.3.9, à ôîðìóëó (6.3.10) âèâåñòè ñàìîñò³éíî.<br />
Çàóâàæåííÿ 1. ßê áóëî îá³öÿíî, â öüîìó ïóíêò³ ìè<br />
íàâåëè çì³ñòîâí³ ïðèêëàäè, ÿê³ ïîêàçóþòü, ùî ôóíêö³¿ sin x,<br />
tg x, arcsin x, arctg x, ln(1+x) åêâ³âàëåíòí³ ôóíêö³¿ õ, êîëè õ<br />
ïðÿìóº äî íóëÿ, òîáòî sin x ~ x, tg x ~ x, arcsin x ~ x,<br />
arctg x ~ x ïðè x → 0.<br />
Çàóâàæåííÿ 2. ßê áóëî âæå ñêàçàíî, íàâåäåí³ âàæëèâ³<br />
ãðàíèö³ åôåêòèâíî âèêîðèñòîâóþòüñÿ äëÿ îá÷èñëåííÿ ð³çíèõ<br />
òèï³â ãðàíèöü. Öå ä³éñíî òàê, ³ ÷èòà÷ ó öüîìó âïåâíèòüñÿ.<br />
Âàæëèâ³ñòü ãðàíèöü çíà÷íî çðîñòå, ÿêùî ìè ï³äêðåñëèìî<br />
¿õ åôåêòèâí³ñòü ïðè çíàõîäæåíí³ ïîõ³äíèõ â³ä<br />
áàãàòüîõ åëåìåíòàðíèõ ôóíêö³é. Ó áàãàòüîõ âèïàäêàõ öå<br />
çðîáèòè ïðîñòî áóëî á íåìîæëèâî. Ïîíÿòòÿ ïîõ³äíî¿ ³ íàâåäåí³<br />
ôàêòè ðîçãëÿíåìî ó íàñòóïí³é òåì³.<br />
ÂÏÐÀÂÈ<br />
6.14. Äîâåñòè çà îçíà÷åííÿì Ãåéíå àáî íà ìîâ³ ïîñë³äîâíîñò³,<br />
ùî limc<br />
= c, äå ñ º ñòàëà.<br />
x→x0<br />
6.15. Äîâåñòè çà îçíà÷åííÿì Ãåéíå àáî íà ìîâ³ ïîñë³äîâíîñò³,<br />
ùî limx<br />
= x .<br />
x→x0<br />
6.16. Äîâåñòè, ùî<br />
0<br />
limx<br />
k = x , k ∈ N.<br />
x→x0<br />
6.17. Íåõàé<br />
1 2<br />
( ) n n -<br />
p x a x a x a x n -<br />
= + + + ... + a x+<br />
a<br />
k<br />
0<br />
n 0 1 2 n-1<br />
n<br />
ìíîãî÷ëåí ñòåïåíÿ n. Äîâåñòè, ùî<br />
lim p n ( x) = p n ( x ) .<br />
x→x0<br />
0<br />
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