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Ó íàâåäåíîìó ïðèêëàä³ òðåáà â³äçíà÷èòè, ùî ôóíêö³ÿ<br />
ó = õ âèçíà÷åíà ³ â òî÷ö³ õ 0 = 0 ³ â í³é äîð³âíþº íóëþ.<br />
2<br />
x + x<br />
Ïðèêëàä 6.2. 2. y= ; a=− 1, b= 1, x0<br />
= 0.<br />
x<br />
Öåé ïðèêëàä ñóòòºâî â³äð³çíÿºòüñÿ â³ä ïîïåðåäíüîãî.<br />
ßêùî â ïðèêëàä³ 6.2.1 çì³ííà õ ìîãëà ïðèéìàòè çíà÷åííÿ<br />
õ = 0, òî òóò äëÿ çàäàíî¿ ôóíêö³¿ öå çðîáèòè íåìîæëèâî.<br />
Ïðè õ = 0 ìè ìàºìî íåâèçíà÷åí³ñòü âèäó 0 . Ùî öå òàêå,<br />
0<br />
ìè ïîêè ùî íå çíàºìî. Àëå ïðè äîñë³äæåíí³ ïîâåä³íêè<br />
2<br />
x + x<br />
ôóíêö³¿ fx ( ) = ïðè íàáëèæåíí³ õ äî íóëÿ íå îáîâ’ÿçêîâî<br />
äàâàòè çíà÷åííÿ õ = 0. Ìîæíà çàäàâàòè ïîñë³äîâí³ñòü<br />
x<br />
{x n }, åëåìåíòè ÿêî¿ ïðÿìóþòü äî íóëÿ, àëå â³äì³íí³ â³ä íóëÿ<br />
( ≠0,<br />
∀ ∈ )<br />
xn<br />
n N .<br />
³äïîâ³äí³ çíà÷åííÿ ôóíêö³¿ òàê³:<br />
y<br />
n<br />
( x + 1)<br />
xn<br />
n<br />
= = xn<br />
+ 1 .<br />
x<br />
n<br />
Òóò ñêîðî÷åííÿ ìîæíà çä³éñíèòè, îñê³ëüêè xn<br />
≠ 0, ∀n∈ N .<br />
Òåïåð ö³ëêîì î÷åâèäíî, ùî ÿêùî lim x n<br />
= 0 , òî lim y n<br />
= 1.<br />
n→∞<br />
n→∞<br />
1<br />
Ïðèêëàä 6.2.3. y = ; a =− 1, b = 1, x<br />
0<br />
= 0 .<br />
x<br />
ßê ³ â ïðèêëàä³ 6.2.2, ó òî÷ö³ õ = 0 ôóíêö³ÿ y = 1<br />
x<br />
íåâèçíà÷åíà,<br />
³ òîìó ïðè äîñë³äæåíí³ ïîâåä³íêè ö³º¿ ôóíêö³¿ ïðè<br />
ïðÿìóâàíí³ x äî íóëÿ òðåáà áðàòè çíà÷åííÿ õ â³äì³íí³ â³ä<br />
x ≠0,<br />
∀n∈N ³ òàê³, ùî lim x n<br />
= 0 . Çíà÷åííÿ ôóíêö³¿<br />
íóëÿ ( )<br />
n<br />
n→∞<br />
îá÷èñëþþòüñÿ çà äîïîìîãîþ ôîðìóëè<br />
1<br />
yn<br />
= , à öå îçíà÷àº<br />
x<br />
n<br />
çã³äíî ç òåîðåìîþ 5.1.1, ùî ïîñë³äîâí³ñòü {y n } º íåñê³í÷åííî<br />
âåëèêîþ ³ òîìó lim y n<br />
=∞.<br />
n→∞<br />
Ïðèêëàä 6.2.4. y = sin 1 ; a = − 1, b = 1, x0<br />
= 0 . ßê ³ â ïîïåðåäí³õ<br />
âèïàäêàõ, ó òî÷ö³ õ = 0 ôóíêö³ÿ y = sin íå âè-<br />
x<br />
1<br />
x<br />
çíà÷åíà ³ òîìó çíîâó æ òàêè ïðè äîñë³äæåíí³ ¿¿ ïîâåä³íêè<br />
òðåáà áðàòè çíà÷åííÿ õ, â³äì³íí³ â³ä íóëÿ ( x 0, n )<br />
n<br />
≠ ∀ ∈N ³<br />
òàê³, ùî lim x n<br />
= 0 . Äàë³ â³çüìåìî äâ³ ïîñë³äîâíîñò³, ÿê³ çàäîâîëüíÿþòü<br />
îñòàííþ<br />
n→∞<br />
óìîâó.<br />
1)<br />
2)<br />
1<br />
{ xn}, xn<br />
= , n∈N .<br />
πn<br />
1<br />
{ xn}, xn<br />
= , n∈N .<br />
π + 2 π n<br />
2<br />
1<br />
³äïîâ³äí³ çíà÷åííÿ ôóíêö³¿ yn<br />
= sin â ïåðøîìó âèïàä-<br />
xn<br />
, à â äðóãîìó âèïàäêó âîíè<br />
⎛<br />
äîð³âíþþòü<br />
⎛π ⎞ ⎞<br />
1⎜sin⎜<br />
+ 2π n⎟=<br />
1⎟. Îòæå, â íàâåäåíîìó ïðèêëàä³<br />
⎝ ⎝2<br />
⎠ ⎠<br />
êó äîð³âíþþòü íóëþ ( sin π n = 0)<br />
1<br />
ïîâåä³íêà ôóíêö³¿ y = sin ïðè ïðÿìóâàíí³ õ äî íóëÿ íå<br />
x<br />
îäíîçíà÷íà. Öåé ïðèêëàä ïîêàçóº, ùî íå ³ñíóº òàêå ºäèíå<br />
÷èñëî, äî ÿêîãî íàáëèæàþòüñÿ çíà÷åííÿ ôóíêö³¿ (ïîñë³äîâí³ñòü<br />
{y n } íå çá³ãàºòüñÿ).<br />
Ïðèêëàä 6.2.5. y= x⋅ sin 1 ; a=− 1, b= 1, x0<br />
= 0 .<br />
x<br />
170 171
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