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7.14.9.<br />
=<br />
ln x 1/ x<br />
=<br />
1/ x 1<br />
−<br />
= − =<br />
3 4<br />
3 x<br />
3 3<br />
lim xln x lim lim 3 lim x 0<br />
x→+ 0 x→+ 0 3 x→+ 0 x→+<br />
0<br />
sin ϕ−1 cos ϕ<br />
lim tgϕ−secϕ = lim = lim = 0 ;<br />
cos ϕ −sin<br />
ϕ<br />
7.14.10. ( )<br />
ϕ→π/2 ϕ→π/2 ϕ→π/2<br />
⎛ 1 x ⎞ x−1−xlnx −lnx<br />
lim⎜<br />
− ⎟= lim = lim<br />
=<br />
ln x x 1 x 1 ln x ln x x 1 / x<br />
7.14.11. x→1 x→1 ( )<br />
x→1<br />
⎝ − ⎠ − + ( − )<br />
xln x 1+<br />
ln x 1<br />
=− lim<br />
=− lim =− .<br />
x→1 xln x+ x− 1 x→1<br />
2 + ln x 2<br />
Òóò ïðàâèëî Ëîï³òàëÿ çàñòîñîâàíî äâ³÷³.<br />
7.14.4. Ðîçêðèòòÿ íåâèçíà÷åíîñòåé 1 ∞ , ∞ 0 òà 0 0<br />
³äçíà÷èìî:<br />
1) ùî íåâèçíà÷åí³ñòü 1 ∞ ÿâëÿº ñîáîþ ãðàíèöþ ñòåïåíåâî-ïîêàçíèêîâî¿<br />
ôóíêö³¿, îñíîâà ÿêî¿ ïðÿìóº äî îäèíèö³, à<br />
ïîêàçíèê ïðÿìóº äî íåñê³í÷åííîñò³;<br />
2) ùî íåâèçíà÷åí³ñòü ∞ 0 ÿâëÿº ñîáîþ ãðàíèöþ ñòåïåíåâî-ïîêàçíèêîâî¿<br />
ôóíêö³¿, îñíîâà ÿêî¿ ïðÿìóº äî íåñê³í÷åííîñò³,<br />
à ïîêàçíèê ïðÿìóº äî íóëÿ;<br />
3) ùî íåâèçíà÷åí³ñòü 0 0 òàêîæ ÿâëÿº ñîáîþ ãðàíèöþ ñòåïåíåâî-ïîêàçíèêîâî¿<br />
ôóíêö³¿, îñíîâà ³ ïîêàçíèê ÿêî¿ ïðÿìóþòü<br />
äî íóëÿ.<br />
Ö³ âèïàäêè çíàõîäæåííÿ ãðàíèö³ ôóíêö³¿ çâîäÿòüñÿ äî<br />
âèïàäêó 0 ⋅∞, à ïîò³ì äî âèïàäêó 0/0 àáî ∞/∞ òàêèì øëÿõîì:<br />
ôóíêö³ÿ ëîãàðèôìóºòüñÿ ³ ñïî÷àòêó çíàõîäèòüñÿ ãðàíèöÿ<br />
¿¿ ëîãàðèôìà, à ïîò³ì ïî çíàéäåí³é ãðàíèö³ ëîãàðèôìà<br />
çíàõîäèòüñÿ ãðàíèöÿ ñàìî¿ ôóíêö³¿.<br />
Ïðèêëàäè 7.14.12 — 7.14.14. Çíàéòè ãðàíèö³:<br />
7.14.12 lim ( tg ) tg 2 x<br />
x ; 7.14.13 lim ( ln x) 1/<br />
x→π/4<br />
x→+∞<br />
x<br />
;<br />
;<br />
7.14.14<br />
6<br />
1 2lnx<br />
lim x +<br />
x→+<br />
0<br />
.<br />
Ðîçâ’ÿçàííÿ.<br />
7.14.12. Ñïî÷àòêó âñòàíîâëþºìî, ùî ìຠì³ñöå âèïàäîê<br />
íåâèçíà÷åíîñò³ 1 ∞ . Ïîò³ì ëîãàðèôìóºìî ôóíêö³þ ³ çíàõîäèìî<br />
ãðàíèöþ ¿¿ ëîãàðèôìà:<br />
tg2x<br />
tg2x<br />
ln tgx<br />
a = lim ( tg x) , ln a = ln lim ( tgx)<br />
= lim tg2xln tgx<br />
= lim<br />
x→π/4 x→π/4 x→π/4 x→π/4<br />
ctg2 x ,<br />
çâåëîñÿ äî âèïàäêó íåâèçíà÷åíîñò³ 0/0. Çàñòîñóâàâøè ïðàâèëî<br />
Ëîï³òàëÿ, îòðèìàºìî:<br />
2<br />
⎡sec<br />
x<br />
2<br />
⎤<br />
ln a = lim ⎢ : ( − 2cos ec 2x)<br />
⎥ = −1<br />
x→π/4<br />
⎣ tg x<br />
.<br />
⎦<br />
Çâ³äñè à = å -1 .<br />
7.14.13. Ç’ÿñóâàâøè, ùî ìຠì³ñöå âèïàäîê íåâèçíà÷åíîñò³<br />
∞ 0 , ðîáèìî ïåðåòâîðåííÿ:<br />
( x)<br />
1/ x<br />
1/ x ln ln<br />
a = lim ( ln x) , ln a = lim ln ( ln x)<br />
= lim .<br />
x→+∞ x→+∞ x→+∞<br />
x<br />
Îòæå, îòðèìàëè âèïàäîê íåâèçíà÷åíîñò³ ∞/∞. Çàñòîñîâóºìî<br />
ïðàâèëî Ëîï³òàëÿ:<br />
⎛ 1 ⎞<br />
a = = ⇒ a = e =<br />
⎝ xln<br />
x ⎠<br />
0<br />
ln lim ⎜ :1⎟<br />
0 1<br />
x→+∞<br />
7.14.14. Ïåðåêîíàâøèñü, ùî ìຠì³ñöå âèïàäîê íåâèçíà-<br />
÷åíîñò³ 0 0 , ïåðåòâîðþºìî:<br />
6 6<br />
6lnx<br />
1+ 2lnx<br />
1+<br />
2lnx<br />
a = lim x , ln a = lim ln x = lim .<br />
x→+ 0 x→+ 0 x→+<br />
0 1 + 2ln x<br />
Òàêèì ÷èíîì, îòðèìàëè âèïàäîê íåâèçíà÷åíîñò³ ∞/∞. Çàñòîñîâóºìî<br />
ïðàâèëî Ëîï³òàëÿ:<br />
.<br />
x→+<br />
0<br />
( )<br />
3<br />
ln a = lim 6 / x: 2 / x = 6 / 2 = 3 ⇒ a = e .<br />
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