x + 1 x3 - 4x2 - 5x x3 - 27
a) lim , b) lim , c) lim ,
x2 x5 - 6x 5 x
x3 - 3
x2 + 1 x2
"+

2x3 + 5x2 - x + 5 (x - 1) 2 - x 1 3
d) lim , e) lim , f) lim - ,
x" - 2x - 2 x2 - 1 1
x1 x1 - x 1 - x3
4x3
x3 - 8 x3 + 3x2 + 2x
g) lim , h) lim ,
x2 - 2 x2
x-2 - x - 6
x
" " "
"
x - 1 - 2 x + 13 - 2 x + 1 x - 5
a) lim , b) lim , c) lim ,
x5 - 5 x2 - 9 x
x3 x25 - 25
x
" " "
tgx x2 + 1 - 1 x2 + 1 - x + 1
d) lim " , e) lim " , f) lim " ,
x0 - 1 + tgx
x0 x0
1
x2 + 25 - 5 1 - x + 1
"
2x
g) lim ( x2 + 1 + x), h) lim " .
x-" x-"
x2 + 1
sin t
lim = 1,
t0
t
sin 5x sin(x - 2) 3x
a) lim , b) lim , c) lim ,
x0 x2 - 4 5 sin 7x
x0
x
" "2x
1 - 1 - x 3x + 4 - 2 sin 3x
d) lim , e) lim , f) lim ,
x0 x0 x0
sin 4x sin 5x 4x
4x sin 2x tg2x
g) lim , h) lim , i) lim ,
x0 x0 x0
3 sin 2x sin 3x sin 3x
"
1 1 - cos x x2 - 4
j) lim x sin , k) lim , l) lim .
x" x0 x-2
x sin x arctg(x + 2)
cos 2x tg2x 1 - x
a) lim , b) lim , c) lim arcsin ,
Ä„
x0 x"
x - cos x tgx 1 + x
sin x
4
arcsin(x + 3) x + 1 arctgx
d) lim , e) lim arctg , f) lim ,
x-3 x" x0
x2 + 3x x + 2 x
arcsin(x + 2) sin 5x - sin 3x
g) lim , h) lim , i) lim ln(xctgx),
x-2 x0
x2 + 2x sin 3x
2x-1 x0
1 x + 1
x
j) lim(1 + 3x) , k) lim , l) lim x(ln(x + 1) - ln x).
x0 x" - 2
x"
x
x2-2x-8
limx4 x2-9x+20
limx1 xn-1, n " N
x-1
" "
"
limx0 x2+1- x+1
1- x+1
4x
limx0 3 sin 2x
8-x
limx8 sin( Ä„x)
1
8
limxĄ 1+cos x
sin2 x
x0
sin Ä…x
x0 = 0
x
x2
x2
x0 = +"
2+x2
1

x2 x2
x0 = 0
2+x2
arc tg x
x0 = 0
x
arc sin x
x0 = 0
x
1
sin x0 = 0
x
1
x sin x0 = 0
x
1 1
a) f(x) = , x0 = 1, b) f(x) = , x0 = 1,
(1 - x) (1 - x)2
1 1
x
c) f(x) = , x0 = 1, -1, d) f(x) = 2 , x0 = 0,
1 - x2
x x
e) f(x) = , x0 = 3, f) f(x) = , x0 = 3,
(3 - x)2 (3 - x)3
1 1
1-x
g) f(x) = 2 , x0 = 1, h) f(x) = , x0 = 2, -2,
x2 - 4
1
1
x2-4 x
i) f(x) = 4 , x0 = 2, -2, j) f(x) = e- , x0 = 0,
1
|2 - x|
1-x3
k) f(x) = e , x0 = 1, l) f(x) = , x0 = 2,
x - 2
|x - 1|
m) f(x) = 2ctgx, x0 = 0, n) f(x) = + x, x0 = 1,
x - 1
1
o) f(x) = arctg , x0 = 1.
1 - x
x0
f(x) = x2 + x x0 = 10
xn
x e" 0 f(x) = limn+" 1+xn x0 = 1
1
x
f(x) = xe x0 = 0
1
x-a
f(x) = 2 x0 = a a " R
x
"
f(x) = x0 = 0
| sin x|

x2-4 sin x
x = -2 x = 0

x+2 |x|
a) f(x) = , b) f(x) = ,
-4 x = -2 1 x = 0

ex-1
x = 0 2x + 3 x d" 0

x
c) f(x) = , d) f(x) = ,
1 x = 0 (x - 2)2 x > 0

1
1-cos x
x
2 x = 0 x = 0

x2
e) f(x) = , f) f(x) = ,
1
1 x = 0 x = 0
2
"
Ä„ 1
2ctgx x " (-Ä„ , ) \ {0} xarctgx x > 0
2 2
g) f(x) = , h) f(x) = .
0 x = 0 0 x = 0
f : R - R f(x) = x2
1
f : R \ {0} - R f(x) =
x
"
f : R+ - R f(x) = x
f : R - R f(x) = |x|

1 x " Q
f : R - R f(x) =
0 x " Q

0 x " Q
f : R - R f(x) =
x x " Q

1 x = 0
f : R - R f(x) =
sin x
x = 0

|x|
x2-x3
f : R \ {1} - R f(x) =
|x-1|
f : R - R f(x) = x - -x
a b

0 , x = 0
f(x) =
1
x sin , x = 0

x
Å„Å‚(x-1)
2
òÅ‚ , |x| = 1

x2-1
f(x) =
a , x = -1
ół
b , x = 1

x , |x| d" 1
f(x) =
x2 + ax + b , |x| > 1
ln(5+x)-ln 5
"
1+x-1
x = 0 x = 0

x x
a) f(x) = , b) f(x) = ,
a x = 0 a x = 0

1
x
x

2 + 1 x < 0 x = 0
|x|
c) f(x) = , d) f(x) = ,
3x2 + a x e" 0 a x = 0
Å„Å‚
2

"
ôÅ‚
òÅ‚x + b x > 2
sin22x
x = 0

2x
e) f(x) = - 1 x = 2 , f) f(x) = .
2a
ôÅ‚x +x-6
a x = 0
2
ół
x < 2
2-x

"C "x,x "R |f(x) - f(x )| d" C|x - x |
"
f : [0; 1) - R f(x) = x
f : R - R f(x) = x2
f : [-a; a] - R f(x) = x2 a > 0
1
f : (0; 1] - R f(x) =
x
1
f : (0; 1] - R f(x) = sin
x
"
f : [0; 1] - R f(x) = x
"
f : [1; ") - R f(x) = x
"
limx0 x = 0
"
"
limxx x = x0
0
lim supx0 f(x) lim infx0 f(x)
1
f(x) = sin
x
1
f(x) = sin2 1
x2 x


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