f(x)
g(x)
f (x)
0 "
0 " g (x)
f(x)
g(x)
ex - e-x 0 H ex + e-x
lim = = lim = 2
x0 x0
x 0 1
" "

1
ln x 0 H 1 x2 - 1 x2 - 1
x
"
lim = = lim = lim · = lim = 0
x1+ 0 x1+ "2x x1+ x x x1+ x2
x2 - 1
2 x2-1
ex - e-x 0 H ex + e-x
lim = = lim = 2
x0 x0
sin x 0 cos x

sin 3x 0 H 3 cos 3x 3
lim = = lim =
x0 x0
sin 5x 0 5 cos 5x 5
(ex - e-x)2 0 H 2(ex - e-x)(ex + e-x) e2x - e-2x
lim = = lim = 2 lim =
x0 x0 x0
x2 cos x 0 2x cos x + x2 sin x 2x cos x + x2 sin x

0 H 2e2x + 2e-2x
= = 2 lim = 4
x0
0 2 cos x - 2x sin x + 2x sin x + x2 cos x

sin x - x cos x 0 H cos x - (cos x - x sin x) x sin x 1 sin x 1
lim = = lim = lim = lim =
x0 x0 x0 x0
x3 0 3x2 3x2 3 x 3

1
"
tg x - 1 0 H 2
cos2 x
"
lim = = lim = = 2
Ä„ Ä„
x - cos x 0 cos x + sin x
x
sin x
2
4 4

1 1 1 1 1 1
x2 sin sin sin 0 H - cos - cos 1
x x x x2 x x
lim = lim = lim = = lim = lim · =
2x-1 2 1 2 2 2 2
x" x" x" x" x"
2x - 1 - 0 - + - + x2
x2 x x2 x2 x3 x2 x3
1
- cos -1 1
x
= lim = =
2
x"
-2 + -2 2
x
ln(ln x)
lim
x"
x
e2x - 1
lim
x0
ln(1 + 2x)

1 x
lim -
x1
ln x ln x
-1
ln(1 - x) " H -(1 - x)2
1-x
lim (1 - x) ln(1 - x) = [0 "] = lim = = lim = lim =
x1- x1- 1 " x1- 1 x1- 1 - x
1-x (1-x)2
= - lim (1 - x) = 0
x1-

1 1
lim -
x0
x sin x
arcsin x
lim
x0
x
arctan x
lim
x0
sin x

1
x
lim x = "0 eln f(x) = f(x)
x"
1
1 1 ln x
x
ln x
x x x
lim x = lim eln x = lim e = lim e = eA
x" x" x" x"

1
ln x " H x 1
x
A = lim = = lim = 0 lim x = eA = e0 = 1
x" x" x"
x " 1
ln x
1-x
lim x1/(1-x) = [1"] lim eln x1/(1-x) = lim e = eA
x1 x1 x1

ln x 0 H
A = lim = = ...
x1
1 - x 0
lim (tg x)tg x
x0+
"
lim x ln x
x0+
lim (sin x)x
x0+
2

arctan x 1/x
lim
x0
x