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Ïðèêëàä 10.2.2. Ïîêàæåìî, ùî ôóíêö³ÿ<br />
2<br />
2x y<br />
2 2<br />
íà ïî÷à-<br />
x + y<br />
òêó êîîðäèíàò ìຠãðàíèöþ, ÿêà äîð³âíþº íóëþ.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Âèêîðèñòîâóºìî î÷åâèäíó íåð³âí³ñòü<br />
( x y) 2 0<br />
− ≥ .<br />
Çâ³äñè âèïëèâàº, ùî<br />
2 2<br />
2 xy≤ x + y.<br />
Îòæå,<br />
2<br />
2x y<br />
x<br />
2 2<br />
x + y ≤ .<br />
Îñê³ëüêè ïðè x → 0 ïðàâà ÷àñòèíà îñòàííüî¿ íåð³âíîñò³<br />
ïðÿìóº äî íóëÿ, òî ³ ë³âà ÷àñòèíà ¿¿ ïðÿìóº äî íóëÿ, òîáòî<br />
lim<br />
2<br />
2xy<br />
lim = 0<br />
2 2<br />
x + y .<br />
x→0<br />
y →0<br />
Ïðèêëàä 10.2.3. Îá÷èñëèòè<br />
Ðîçâ’ÿçàííÿ<br />
lim<br />
x<br />
2<br />
+<br />
2<br />
+ y + − .<br />
x y1 x →0<br />
2 2<br />
1<br />
y →0<br />
2 2 2 2<br />
2<br />
x + y ⎡ x + y = ρ ⎤<br />
ρ<br />
= ⎢<br />
⎥ = lim<br />
=<br />
+ + 1 −1 ⎢x<br />
+ y → 0 ⇒ ρ → 0⎥<br />
ρ + 1 −1<br />
x→0 2 2 2 2<br />
ρ→0<br />
2<br />
y→0<br />
x y<br />
⎣<br />
⎦<br />
( 1 1)<br />
2 2<br />
ρ ρ + +<br />
= lim<br />
= 2<br />
2<br />
.<br />
ρ→0<br />
ρ<br />
Íà çàê³í÷åííÿ ïàðàãðàôà ðîçãëÿíåìî ïîíÿòòÿ íåïåðåðâíîñò³<br />
ôóíêö³¿ äâîõ çì³ííèõ.<br />
Îçíà÷åííÿ 10.2.2. Ôóíêö³ÿ z = f(x, y) íàçèâàºòüñÿ íåïåðåðâíîþ<br />
â òî÷ö³ M 0 (x 0 , y 0 ), ÿêùî<br />
( , ) = ( , )<br />
limf x y f x y<br />
x→x0<br />
y→y0<br />
0 0<br />
.<br />
Îçíà÷åííÿ 10.2.3. Ôóíêö³ÿ z = f(x, y) =f(M) íàçèâàºòüñÿ<br />
íåïåðåðâíîþ ó â³äêðèò³é ÷è çàìêíóò³é îáëàñò³, ÿêùî âîíà<br />
íåïåðåðâíà â êîæí³é òî÷ö³ ö³º¿ îáëàñò³.<br />
Çàóâàæåííÿ 1. Ôóíêö³ÿ z = f(M) ââàæàºòüñÿ íåïåðåðâíîþ<br />
â ãðàíè÷í³é òî÷ö³ M 0 , ÿêùî lim fM ( ) fM ( )<br />
M→M0<br />
= , êîëè<br />
òî÷êà M ïðÿìóº äî òî÷êè M 0 óçäîâæ áóäü-ÿêîãî øëÿõó, ùî<br />
íàëåæèòü äàí³é îáëàñò³.<br />
Ðàí³øå áóëè ðîçãëÿíóò³ âëàñòèâîñò³ ôóíêö³é îäí³º¿ çì³ííî¿,<br />
íåïåðåðâíî¿ íà ñåãìåíò³. Àíàëîã³÷í³ âëàñòèâîñò³ ìàº<br />
ôóíêö³ÿ äâîõ çì³ííèõ.<br />
Ìຠì³ñöå<br />
Òåîðåìà 10.2.2. ßêùî ôóíêö³ÿ z = f(x, y) íåïåðåðâíà â<br />
îáìåæåí³é çàìêíóò³é îáëàñò³, òî âîíà â ö³é îáëàñò³<br />
1) îáìåæåíà: f( x,<br />
y)<br />
≤ M, M >0;<br />
2) ìຠíàéìåíøå çíà÷åííÿ m ³ íàéá³ëüøå çíà÷åííÿ M;<br />
3) ïðèéìຠõî÷à á â îäí³é òî÷ö³ îáëàñò³ áóäü-ÿêå ÷èñåëüíå<br />
çíà÷åííÿ, óêëàäåíå ì³æ m ³ M;<br />
4) ôóíêö³ÿ äîð³âíþº íóëþ â òî÷ö³ îáëàñò³, ÿêùî ³ñíóþòü<br />
òî÷êè, ó ÿêèõ ôóíêö³ÿ ïðèéìຠçíà÷åííÿ ð³çíèõ çíàê³â.<br />
2 2<br />
Ïðèêëàä 10.2.4. Äîâåñòè, ùî ôóíêö³ÿ z = sin( x + y ) íåïåðåðâíà<br />
íà ïî÷àòêó êîîðäèíàò.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Äëÿ êîæíîãî ε > 0 ³ñíóº òàêå δ, ùî ÿê<br />
2 2<br />
2 2<br />
ò³ëüêè ρ ( M,0) = x + y
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