"
"
"
 I x " I f
f (x) = 0 I
f (x) > 0 I
f (x) < 0 I
x
f(x) =
1 + x2
f(x) = sin x + cos x
f(x) = 3x5 + 5x3
f(x) = x + cos x
0
0
f g
lim f(x) = lim g(x) = 0 g(x) = 0

xx0 xx0
f (x)
lim
xx0
g (x)
f(x) f (x)
lim = lim .
xx0 xx0
g(x) g (x)
-" "
sin 3x
lim
xĄ
sin 5x
x10 - 1
lim
x1 - 1
x3
Ä„ - 2 arctan x

lim
1
x"
ln 1 +
x2
ln cos x
lim
x0+ ln cos 2x
"
"
f g
lim f(x) = lim g(x) = "
xx0 xx0
f (x)
lim
xx0
g (x)
f(x) f (x)
lim = lim .
xx0 xx0
g(x) g (x)
 -" "
ln x
lim
x"
x
tan 3x
lim
Ä„
x tan x
2
x3 - 2x + 1
lim
x"
4x3 + 2
ln sin x
lim
x0+ ln tan x
f 0 "
0 · " f · g =
1
0 "
g
1 1
-
0
g f
" - " f - g =
1
0
fg
1" "0 00 fg = eg ln f 0 · "
" - "
lim x ln x
x0+

1 1
lim -
x0
2x2 2x tan
xx
2
lim arctan x
x"
Ä„
f x0 " Df
"´>0"x"(x f(x) > f(x0).
0-´,x0+´)
f x0 " Df
"´>0"x"(x (f(x) < f(x0).
0-´,x0+´)
 f
f (x0) = 0
f (x) x x0
f x0
f(x) = x3 - 3x2 + 4
g(x) = ex + e-x
ln x
h(x) =
x
f
f (x0) = 0
f (x0) > 0 (f (x0) < 0)
f x0
f(x) = x100 + 2x50
g(x) = (x - 5)ex
(a, b)
"a1
(a, b)
"a1
f (x) > 0 x " (a, b) f (a, b)
f (x) < 0 x " (a, b) f (a, b)
 f(x) = e-x
g(x) = x4
h(x) = sin x
s(x) = arctan x
f x0 (x0, f(x0))
f ´ > 0
f (x0-´, x0) (x0, x0+´)
f
f (x0) = 0
f (x) x x0
f x0
1
f(x) = 4x2 +
x
g(x) = ecos x
h(x) = x2 ln x
"
3
s(x) = 1 - x3