Limes funkcije
10 Limes funkcije
Zadatak 10.3 x 2 7x + 10
5) (x 4) x!5 x 4 5 4 Zadatak 10.4
x 2 2 10x x+ x 2 10x lim x
x 10x = lim
= 10x
x!1
x!1
1 x+ x 2
x 2 x 2 10x 2
x 2 x 2 + 10x
= lim
p 2 = lim p
= 10x
x!1
x+ x
10x
x!1 x+ x 2
10
10x
:x
= lim
10x :x x!1 x 2 10x =
p 2 = lim
x!1 x+ x
1+
10 10
= lim
= lim
x!1
1+ 10 x 2 1+ 1 x
10x
x!1
10 10
= 5:
1+ 1 0 2
Zadatak 10.5
1 1+x 2 1 1+x 2 + 1+x 2 +1 lim
1+x 2
2 x!0 = x x!0 x p 3
2 = lim
1+x 2 2 p
+ 3 1+x 2 +1
1+x 2 3 1 3
= lim
= 1+x 2 + 1+x 2 +1
x!0
x 3 x 3 x+1+2 lim p
= lim p
x!3
x+1 2 x!3 x+1 2 x+1+2
Zadatak 10.8 p
2 x 3 2 x 3 2+ x 3 lim
= lim
x!7
x 2 49 x!7 x 2
49 2+ x 3
= lim
x!7 (x 2
49) 2 + x 3
4 (x 3)
= lim
p (x 2 49) 2 + x 3
x!7 x!7
= lim p
x!4
x+5 3 x!4 x+5 3 x+5+3
(x
16) x+5+3
= lim
2 x!4 =
x+5
(x
4) (x + 4)
x+5+3
= lim
x!4
x+5 9
(x
4) (x + 4)
x+5+3
= lim
x!4
= lim (x + 4)
x + 5 + 3 = 8 6 = 48:
x!4
Zadatak 10.10
x 3 27 x 3 3 3 12 x+3 lim p
= lim p
x!3
12 x 3 x!3 12 x 3 12 x+3
(x
3 ) 12 x+3
= lim
x!3
12 2 x 3 2
(x
3) (x 2 + 3x + 9) 12 x+3
= lim
x!3
12 x 9
2 (x p 3) (x + 3x + 9) 12 x+3
= lim
x!3
(x
3) (x 2 + 3x + 9) 12 x+3
= lim
x!3
(x 3)
lim x + 3x + 9
12 x+3=
x!3
Zadatak 10.12 x 2 + 5x + 4 3x
x2 3x+7 8x 3 8x 3x 3 8x 3 x2 3x+7 = lim 1+ 2 =
x! +1 =e x2 =e 3x+7 =
3x+7 3x
x 2x 2 +x+1 x 2 +x+1 lim
2 2 x 2x +x+1 x +x 1
= lim 1+
x!+1
x 2 x+1 2x 2x
= lim 1+
x!+1
x+1
x2 2x 2x x 2x +1 2x
x2 x +1
= lim 1+
x!+1
x 2 x+1
x! =e lim =e +1 x2 x +1 =e 4 :
lim
2x
x +1 2x
4x2
x! +1 x2
Zadatak 10.14 sin 7x
x+4+2 lim p
x+4 2 x!0 x+4 2 x+4+2
p sin 3x x+4+2
p sin 3x x+4+2
p sin 3x x+4+2
= lim
x!0
p x sin 3x x+4+2
Zadatak 10.17 p
x+1 1 x+1 1 x+1+1 lim
x+1 1 x+1+1
= lim
x!0
sin 2x x+1+1 p
x+1 2 1 2
= lim
x!0 sin 2x x+1+1 x!0 sin 2x x+1+1
= lim
x!0 sin 2x x+1+1 x
= lim
x!0 sin 2x x+1+1 x
x+9+3 lim p
x+9 3 x!0 x+9 3 x+9+3
p sin 4x x+9+3
p sin 4x x+9+3
p sin 4x x+9+3
= lim
x!0
p x sin 4x x+9+3
= lim
x!0
4x sin 4x
= 4 lim
lim
x+9+3=
x!0 4x
x!0
5lim sin 5x 3lim sin 3x 5x 3x = x!0
4 x!0 2 x!0 2 2 2 Zadatak 10.21
Zadatak 10.22 x+1
lim x [ln (x + 1) ln (x 1)] = lim x ln =
x!+1
x!+1
x 1 x+1 x
= lim ln
x!+1
x 1 x+1 x
= ln lim
x!+1
3x+7 3x
3x+7 3x
9x x! lim x2 3x+7
+1 1 x+ 3 7 = ln e 24 x2 = ln e = 24 ln e = 24: Zadatak 10.24
x! lim
lim 2 3x ln (x + 1) + ln (x 4) ln x + 5x + 6 =
x!+1 x!+1
8x 10
x2 +5x+6 3x
= ln e = ln e +1 x2 +5x+6 = ln e 24 = 24: Zadatak 10.25
x! lim
+1 x2 +5x+6 3x
x x+3 +x+1 +1 x+3 +1 lim
x2
x +x+1
x x+3 +x+1 x 2 +1
x 1 +x+1 +1
(x+2) x2
x 1 +2
x 1 +x+1 +1
x! =e lim +1 x2 +3x+2 =e 1 = e: Zadatak 10.26
lim x 1 x2 +x+1
x2 +x+1
=e x! +1 +2 x +1
x t 1 lim
x 1 2 smjena
= 4 1 t= 1
x 2 =) x = 2 + t 5 = lim 1+ =
x!2
x!+1
2t
x ! 2 =) t ! 1
2t 1 1 2 1
= lim 1+
=e 2 :
x!+1
2t
Zadatak 10.27
smjena
lim = 6 1 =) e =t+1
t!0 ln(1+t)
=a
ln(1+t) =a
Zadatak 10.28 Izraµcunati lijevi i desni limes funkcije
3x x 2
f (x) = x 4
u taµcki x = 4:
3x 2 x 3 (4 ") (4 ") L(4) = lim
3x 2 x 2 smjena
3 (4 + ") (4 + ") D(4) = lim
Zadatak 10.29 Izraµcunati lijevi i desni limes funkcije
ln x 2 1 ln x 2 1
D(1) = lim
ln x 2 1 = ln lim
1= 4 x=1+" 5=
x!1 +
x!1
" > 0; " ! 0
2 = ln lim 2 (1 + ") 1 = ln lim 1 + 2" + " 1=
"!0
"!0
= ln lim 2 2" + " =
1:
"!0