Tg x - MENDELU
[Pages:3]Vypoctte n?sleduj?c? limity pomoc? l'Hospitalova
pravidla:
x-1
1. lim
[1]
x1 ln x
2.
lim
x
ln x x
[0]
tg 3x
3. lim
x
2
tg x
[
1 3
]
4.
lim
x0
x
-
arctg x x3
xex + x - 2ex + 2
5. lim
x0
x3
cos 2x - 1
6.
lim
x0
2 sin2
x
+
2x sin 2x
[-
1 3
]
[
1 6
]
[-
1 3
]
7. lim ex - e-x - 2x
[2]
x0 x - sin x
8.
lim
x0
1 - cos x2 x2 sin x2
[
1 2
]
x cotg x - 1
9. lim
x0
x2
[-
1 3
]
Vypoctte n?sleduj?c? limity pomoc? l'Hospitalova pravidla:
Typ
0 0
a
? ?
.
1.
x ln x
lim
x
x2
+
x
+
1
=
||
||
l'H=.p.
lim
x
ln x + 1 2x + 1
||
||
l'H=.p.
lim
x
1 x
2
do=saz. 0
2.
lim
x0
arctg 2x arcsin x
=
||
0 0
||
l'H=.p.
lim
x0
2 1+4x2
1
1-x2
?p=rava
lim
x0
2 1
1 +
- x2 4x2
do=saz. 2
3.
lim x - 1 x1 ln x
=
||
||
l'H=.p.
lim
x1
x
1
do=saz. 1
x
4.
ln x lim x x
=
||
||
l'H=.p.
lim
x
1 x 21 x
=
||
0 0
||
?p=rava
1 2
lim
x
1 x
=0
5.
lim
x 2
tg 3x tg x
=
||
||
l'H=.p.
lim
x
2
3 cos2 3x
1 cos2 x
=
||
||
?p=rava
3
lim
x
2
cos2 x cos2 3x
=
||
0 0
||
l'H=.p.
3
lim
x
2
-2 sin x cos x -6 sin 3x cos 3x
=
||
0 0
||
=
?p=rava
3?
2 6
? lim
x
2
sin x sin 3x
? lim
x
2
cos x cos 3x
?p=rava
- lim
x
2
cos x cos 3x
=
||
0 0
||
l'H=.p.
- lim
x
2
- sin x -3 sin 3x
do=saz.
1 3
5B.
lim
x
2
tg 3x tg x
=
||
||
?p=rava
lim
x
2
sin 3x cos x sin x cos 3x
?p=rava
lim
x
2
sin 3x sin x
? lim
x
2
cos x cos 3x
=
- lim
x
2
cos x cos 3x
=
||
0 0
||
l'H=.p.
=
- lim
x
2
- sin x -3 sin 3x
do=saz.
1 3
6.
lim
x0
x
-
arctg x x3
=
||
0 0
||
l'H=.p.
lim
x0
1
-
1 1+x2
3x2
=
||
0 0
||
?p=rava
lim
x0
3(1
1 + x2)
do=saz.
-
1 3
7.
lim
x0
xex
+
x - 2ex x3
+
2
=
||
0 0
||
l'H=.p.
lim
x0
ex
+ xex + 1 - 2ex 3x2
=
||
0 0
||
l'H=.p.
lim
x0
ex
+ ex
+ xex 6x
- 2ex
?p=rava
= lim xex ?p=rava lim ex do=saz. 1
x0 6x
x0 6
6
8.
lim
x0
xex
+
x - 2e-x x3
+2
=
||
0 0
||
l'H=.p.
lim
x0
ex
+ xex + 1 + 2ex 3x2
do=saz.
4 +0
=
9.
lim
x0
cos 2x - 1 2 sin2 x + 2x sin 2x
=
||
0 0
||
l'H=.p.
lim
x0
-2 sin 2x 4 sin x cos x + 2 sin 2x + 4x cos 2x
=
||
0 0
||
l'H=.p.
=
lim
x0
4 cos x cos x -
-4 cos 2x 4 sin x sin x + 4 cos 2x + 4 cos 2x -
8x sin 2x
do=saz.
-
1 3
10.
lim
x0
ex
- e-x - 2x x - sin x
=
||
0 0
||
l'H=.p.
lim
x0
ex + e-x - 2 1 - cos x
=
||
0 0
||
l'H=.p.
lim
x0
ex - e-x sin x
=
||
0 0
||
l'H=.p.
=
lim
x0
ex + e-x cos x
do=saz. 2
11.
lim
x0
1 - cos x2 x2 sin x2
=
||
0 0
||
l'H=.p.
lim
x0
2x sin x2 2x sin x2 + 2xx2 cos x2
?p=rava
=
lim
x0
sin x2 sin x2 + x2 cos x2
=
||
0 0
||
l'H=.p. lim
x0
2x cos x2
2x cos x2 + 2x cos x2 - 2xx2 sin x2
=
||
0 0
||
?p=rava
=
lim
x0
cos x2
cos x2 + cos x2 - x2 sin x2
do=saz.
1 2
1
2
12.
lim
x0
x cotg x x2
-
1
=
lim
x0
x cos x - sin x x2 sin x
=
||
0 0
||
l'H=.p.
lim
x0
cos x - x sin x - cos x 2x sin x + x2 cos x
?p=rava
=
- lim
x0
2
sin
sin x+
x x
cos
x
||
0 0
||
l'H=.p.
lim
x0
2 cos x
+
cos x cos x
-
x sin x
do=saz.
-
1 3
13.
x2
lim
x
ex2
=
||
||
l'H=.p.
lim
x
2x 2xex2
?p=rava
lim
x
1 ex2
do=saz. 0
14.
lim 1 - x
x1
cotg
x 2
=
||
0 0
||
l'H=.p.
lim
x1
-
2
-1 sin-2
x 2
do=saz.
2
15.
lim
x0
sin x - x x sin x
=
||
0 0
||
l'H=.p.
lim
x0
cos x - 1 sin x + x cos x
=
||
0 0
||
l'H=.p.
lim
x0
- sin x cos x + cos x - x sin x
do=saz.
0
16.
lim
x0
x cos x - sin x x sin x
=
||
0 0
||
l'H=.p.
lim
x0
cos x - x sin x - cos x sin x + x cos x
?p=rava
- lim
x0
-x sin x sin x + x cos x
=
||
0 0
||
l'H=.p.
=
- lim
x0
sin x + x cos x cos x + cos x - x sin x
do=saz.
0
17.
lim
x0+
ex - 1 - x (ex - 1)x
=
||
0 0
||
l'H=.p.
lim
x0+
ex - 1 ex - 1 - xex
=
||
0 0
||
l'H=.p.
lim
x0+
ex ex - xex - ex
do=saz.
1 2
18.
lim
x1
x ln x - x + 1 (x - 1) ln x
=
||
0 0
||
l'H=.p.
lim
x1
ln x + x 1 - 1
x
x-1 + ln x
x
?p=rava
lim
x1
ln x x-1 + ln x
x
?p=rava
lim
x1
(x
x ln x - 1) + x ln x
=
=
||
0 0
||
l'H=.p. lim
x1
1+ 2+
ln x ln x
do=saz.
1 2
arcsin x 19. lim
x0 1 - ex
=
||
0 0
||
l'H=.p.
lim
x0
1
1-x2
-ex
do=saz. -1
20.
lim
xe
ln2
ln x - 1 x + ln x
-
2
=
||
0 0
||
l'H=.p.
lim
xe
1
x
1 ln x + 1
x
x
?p=rava
lim
xe
ln
1 x+
1
do=saz.
1 2
21.
ln x
lim
x1
cotg
x 2
1
=
||
0 0
||
l'H=.p.
lim
x1
x
-
1 sin2
x 2
2
do=saz. - 2
22.
lim
x0+
ln(ex - 1) ln x
=
||
||
l'H=.p.
lim
x0+
1 ex -1
ex
1
x
?p=rava lim
x0+
xex ex -
1
||
0 0
||
l'H=.p.
lim
x0+
ex + xex ex
do=saz. 1
23.
lim
x0
ex - 1 sin 2x
=
||
0 0
||
l'H=.p.
lim
x0
ex 2 cos 2x
do=saz.
1 2
24.
lim
x0
x - sin x sin3 x
=
||
0 0
||
l'H=.p.
lim
x0
3
1- sin2
cos x x cos
x
||
0 0
||
l'H=.p.
lim
x0
sin x 6 sin x cos2 x -
3 sin3
||
0 0
||
x
l'H=.p.
=
lim
x0
6 cos3
x
-
6
?
2
?
cos x sin2 x cos x
-
9 sin2
x cos x
do=saz.
1 6
25.
lim
x3
x2
x-3 - 8x +
15
=
||
0 0
||
l'H=.p.
lim
x3
1 2x -
8
do=saz.
-
1 2
26.
lim
x1
1 + cos x (x - 1)2
=
||
0 0
||
l'H=.p.
lim
x1
- sin x 2(x - 1)
=
||
0 0
||
l'H=.p.
lim
x1
-2
cos x 2
do=saz.
2 2
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