MATEMATIKA 1 skripta studij: Biotehnologija i Prehrambena ... - PBF
MATEMATIKA 1 skripta studij: Biotehnologija i Prehrambena ... - PBF
MATEMATIKA 1 skripta studij: Biotehnologija i Prehrambena ... - PBF
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= 4 lim<br />
x→+∞<br />
b)<br />
= 4 lim<br />
x→−∞<br />
x √<br />
x 2<br />
√<br />
x 2 +2x+3+ √ x 2 −2x+3 √<br />
x 2<br />
lim f(x) = lim<br />
x→−∞ x→−∞<br />
x √<br />
x 2<br />
√<br />
x 2 +2x+3+ √ x 2 −2x+3 √<br />
x 2<br />
= 4 lim<br />
x→+∞<br />
1<br />
√<br />
1+ 2 + 3 +<br />
x x 2<br />
√<br />
1− 2 x + 3<br />
x 2 = 4 2 = 2.<br />
∣ 4x ∣∣∣<br />
√<br />
x2 +2x+3+ √ x 2 −2x+3 = −∞<br />
∞ ∣<br />
= 4 lim<br />
x→−∞<br />
−1<br />
√<br />
1+ 2 + 3 +<br />
x x 2<br />
√<br />
1− 2 x + 3<br />
x 2 = −4<br />
2 = −2.<br />
Zadatak 72 Izračunajte a) lim x→+∞ ( 3√ x 3 +3x 2 −x)<br />
b) lim x→+∞ ( √ x 2 −2x−x) c) lim x→+∞ ( 3√ x 3 +3x 2 − √ x 2 −2x).<br />
Rješenje: a)<br />
√<br />
lim ( 3√ x 3 +3x 2 −x) = |∞−∞| = lim ( 3√ 3 (x3 +3x<br />
x 3 +3x 2 −x)<br />
2 ) 2 +x 3√ x 3 +3x 2 +x 2<br />
√<br />
x→+∞ x→+∞ 3 (x3 +3x 2 ) 2 +x 3√ x 3 +3x 2 +x 2<br />
= lim<br />
x→+∞<br />
= lim<br />
x→+∞<br />
3x 2 ∣ ∣∣<br />
√<br />
3 (x3 +3x 2 ) 2 +x 3√ x 3 +3x 2 +x = ∞<br />
∣<br />
2 ∞<br />
3x 2 | : x 2<br />
√<br />
(x3 +3x 2 ) 2 +x 3√ x 3 +3x 2 +x 2 | : x = 3 lim 2<br />
3<br />
x→+∞<br />
1<br />
= 1.<br />
√(1+ 3 x )2 +<br />
√1+ 3 3 +1 x<br />
b)<br />
√<br />
lim (√ x 2 −2x−x) = |∞−∞| = lim (√ x2 −2x+x<br />
x 2 −2x−x) √<br />
x→+∞ x→+∞ x2 −2x+x<br />
∣ −2x ∣∣∣<br />
= lim √<br />
x→∞ x2 −2x+x = −∞<br />
x 1<br />
∞ ∣ = −2 lim √ = −2 lim = −1.<br />
x→∞ x2 −2x+x x→∞<br />
√1− 2 +1 x<br />
c)<br />
lim<br />
x→+∞ ( 3√ x 3 +3x 2 − √ x 2 −2x) = |∞−∞| = lim<br />
x→+∞ ( 3√ x 3 +3x 2 −x−( √ x 2 −2x−x))<br />
= lim<br />
x→+∞ ( 3√ x 3 +3x 2 −x)− lim<br />
x→∞<br />
( √ x 2 −2x−x) = 1−(−1) = 2.<br />
3<br />
61
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