9
6
x2 - 2x - 1 9 - 6 - 1 2 1
lim = = =
9
x3
9 - 5 4 2
x2 - 5
1
0
(x + 1)ex
lim = 1
1
x0
cos x
x2 - 4 (x - 2)(x + 2)
lim = lim = 1
x2 x2
x - 2 x - 2
x3 - 1 (x - 1)(x2 + x + 1) 3
lim = lim =
x1 x1
x2 - 1 (x + 1)(x - 1) 2

1 2 x + 1 - 2 x - 1 1
lim - = lim = lim =
x1 x1 x1
x - 1 x2 - 1 (x - 1)(x + 1) (x - 1)(x + 1) 2
0
0
0
1 1 1 1
2x3 + x2 - x - 1 (2 + - - ) 2
x3 x x2 x3
lim = lim =
0
0 0
x" x"
5x3 + x2 - x + 1 5
1 1 1 1
(5 + - + )
x3 x x2 x3
0
0 0
1 1 1 1
2x3 + x2 - x - 1 (2x + - - )
x2 x x2 x3
lim = lim = "
0
0
x" x"
5x2 + x + 1
1 1 1
(5 + + )
x2 x x2
0
0 0
0
1 2 1 1 1
2x2 + x2 - x - 1 (x + - - )
x3 3 x x2 x3
lim = lim = 0
0
0 0
x" x"
5x3 + x2 - x + 1
1 1 1 1
(5 + - + )
x3 x x2 x3
2x4 + 1
lim
x-"
5x2 - x
"
5x - 5
lim
x5
x - 5
sin 3x sin 3x 3 3
lim = lim · =
x0 x0
5x 3x 5 5
tg 2x sin 2x 2 2
lim = lim · =
x0 x0
7x 2x cos 2x 7 7

1 - cos x 1 - cos2 x sin x 2 1 1
lim = lim = lim · =
x0 x0
x2 x0 x2(1 + cos x) x 1 + cos x 2
sin 3x
sin 3x 3 3
3x
lim = lim · =
sin 2x
x0 x0
sin 2x 2 2
2x
sin x
tg x - sin x - sin x sin x(1 - cos x) sin x(1 - cos x) 1 + cos x
cos x
lim = lim = lim = lim · =
x0 x0 x0 x0
x3 x3 x3 cos x x3 cos x 1 + cos x
1
sin3 x sin3 x 1 1
lim = lim · =
1
1
x0 x0
x3(cos x + cos2 x) x3 2
cos x + cos2 x
4x 2 2
lim = lim =
1
x0 x0
3 sin 2x 3
3sin 2x
2x
sin x
lim
Ä„
x
x
2
sin x
lim = 0
x"
x
2
x x x
1 1
lim 1 + = lim 1 + = +"
x" x"
x x
2
x x x
1 1
lim 1 + = lim 1 + = 0
x-" x-"
x x
x 3
x
3
3 3
lim 1 + = lim 1 + = e3
x" x"
x x
x
x + 1 x x - 3 + 4 x 4
lim = lim = lim 1 + =
x" x" x"
x - 3 x - 3 x - 3
ëÅ‚ öÅ‚4
x-3 3
4
4 4
íÅ‚ Å‚Å‚
= lim 1 + 1 + = e4
x"
x - 3 x - 3

" "
" "
" x + 5 - x x + 5 + x
"
5
" "
lim ( x + 5 - x) = lim " = lim " = 0
x" x" x"
x + 5 + x x + 5 + x
"
3
lim (x + 1 - x3)
x"
x2 + x
"
lim
x0
1 - x + 1