ą � � ł " � � ś
� Ś � ą � �  �
� ś �  Ą Ł �
� � Ś � � � � &! �
x + y = x y = 0
x � y = x x = 0 y = 1
x = y x + z = y + z

z = 0 x = y x � z = y � z

1 1 1
2 = 1; 2 + 2 = 4; 2 � 2 = 4; -0 = 0; = 1; x = 0, = 0; x = 0 x = 1, = 1.

1 x x
-(-x) = x,
(-x) + (-y) = -(x + y)
x � 0 = 0 � x = 0
1
= x x = 0

1
x
x � y = 0 x = 0 y = 0 R
-x = (-1) � x
x w
= �! x � z = y � w y, z = 0

y z
1 1 1
� = x, y = 0

x y x�y
xz x
= y, z = 0

yz y
xz+yw xz-yw
x w x w
+ = - = y, z = 0

y z yz y z yz
x w xw x w xz
� = y, z = 0 : = y, z, w = 0

y z yz y z yw
 x < y z < w x + z < y + w
x < 0 x = 0

x > 0 -x < 0 x < 0 -x > 0 x > 0 y < 0 x � y < 0 x < 0 y < 0
1
x � y > 0 x > 0 > 0
x
x < y z < 0 y � z < x � z
x y y x x = y x = 0 x � x = 0

x 0 y 0 x � y 0
1 1
x y z w x + z y + w 0 < x < y 0 < <
y x
A = {x " R : x d" 1}, B = {x " R : x > 1}
A = {1}, B = {2, 3}
"
x � x = 2
-|x| x |x|
a " R
|x| < a -a < x < a
|x| > a x < -a x > a
a " R
|x| a -a x a
|x| a x d" -a x a
|x|
|x| 0 |x| = y = 0 |x - y| ||x| - |y||

y |y|
|x||y| = |xy| |x + y| |x| + |y|
3
|x| - |x - 1| = 2 |x - 1| + |x + 1| = x + |x + |x + 1|| = 3
2
|x|
x + |2 - 3x| = 4 |x + 2| + |x - 1| = 3 x + = |x|
x
|x - 1| < 2x + 1 |5x - |x + 3|| e" 3 ||x - 1| + x| e" 2x + 1
|x+1| |x| |x-1|
|x - 2| < x + 2 |x| + 2x e" 2
-3 d" + + d" 3
x+1 x x-1
||x - 2| - 3x| < 3 |x + 2| + 1 < x
|1 - 3x| - |x + 2| d" 1 |2x + 1| + 1 e" x
 E = {x " R : x > 3} E
E �" R
sup E = max E
E �" R
inf E = min E
E �" R
inf E = -2 sup E = 5 inf E = sup E
"
(a, b max(a, b = sup(a, b = b inf(a, b = a
min(a, b
E �" R inf(-E) = - sup E
-E = {x " R : -x " E}
E �" R
inf E sup E
inf E = sup E E
E, E" �" R E �" E" sup E sup E"
E, E" �" R x " E
y " E" x y sup E sup E"
E �" R - inf E = sup(-E)
E, E" �" R E �" E" inf E" inf E
E �" R k > 0 A = {ka : a " E}
sup A = k � sup E
E �" R k < 0 B = {ka : a " E}
inf B = k � sup E
E �" R k " R C = {k - a : a " E}
sup C = k - inf E
E �" R M
E sup E d" M
E �" R m
E m d" inf E
E, E" �" R E + E" = {x " R :
"
"y"E"z"E x = y + z} sup(E + E") = sup E + sup E"
E, E" �" R+ E � E" = {x " R :
"
"y"E"z"E x = yz} sup(E � E") = sup E sup E"
E, E" �" R inf(E + E") =
inf E + inf E"
 E, E" �" R+ inf(E � E") = inf E inf E"
E, E" �" R E *" E"
sup(E *" E") = max{sup E, sup E"}
E, E" �" R E *" E"
inf(E *" E") = min{inf E, inf E"}
A = {x " R : x � x - 5x + 6 < 0} D = {x � x - x : x " R}
B = {x " R : |2x - 5| < 3} E = {|x - 1| + 3 : x " R, -2 < x < 4}
x
C = {x " R : 3 < |x - 1| 4} F = {1+|x| : x " R}

n + 1 n n n n!
"n"N "1 k n = + , 0! = 1, = .
k k k - 1 k (n - k)!k!

n
"n"N "0 k n " N.
k
a fa : N - R
fa(1) = a
fa(n + 1) = fa(n) � a n " N
n, m " N
fa(n + m) = fa(n) � fa(m).
an = fa(n)
n-1
"a,b"R "n"N (a � b)n = an � bn "a,b"R"n"N an-bn = (a-b) an-1-kbk
k=0
"a"R "n,m"N an+m = an � am
"0 "a,b"R "n,m"N (an)m = anm
n
n
"a,b"R "n"N (a + b)n = an-kbk "a>1 "n,m"N,nk=0 k
n(n+1)
"n"N 1 + 2 + � � � + n = "n"N 34n+2+1 10
2
1 1 1 n
"n"N 1�2 + + � � � + =
2�3 n(n+1) n+1
"n"N 34n - 1 5
n(n+1)(2n+1)
"n"N 12 + 22 + � � � + n2 =
6
"n"N n3 - n 6
"n"N 13 + 23 + � � � + n3 = (1 + 2 + � � � + n)2
"n"N 10n - 4 6 "n"N n3 + 2n 3
4
"n"N 2n 2n "n 2 2n
n! n
1 3 2n-1 2 1
"n"N 3n 3n
"n"N 2 � � � � <
4 2n 2n+1
"n 3 3n > n2n
3 5 2n+1 2 9
"n"N 2 � � � � d" n
4 2n 4
"n"N (2n)! < 22n(n!)2
2
4 2n
(2n)!
"n"N 2 � � � � d" 4n
1 3 2n-1
"ne"2 2(n!) <
n+1
"n"N n! e" 2n-1 "n 3 nn+1 > (n + 1)n
"n 2 y1 > 0, . . . , yn > 0 y1 + y2 + � � � + yn = n y1 � y2 � � � yn 1
"n 2 x1 � x2 � � � xn = 1 x1 > 0, x2 > 0, . . . , xn > 0 x1 + x2 + � � � xn n
x1+x2+���+xn n
"n 2 x1 0, x2 0, � � � , xn 0 x1 � x2 � � � xn
n
xn-1 xn
x1 x2
"n 2 x1 > 0, x2 > 0, . . . , xn > 0 + + � � � + + n
x2 x3 xn x1
n+1 n
"n 2 n!
2
xk, yk " R k = 1, . . . , n
n n n
2
2
xkyk x2 yk .
k
k=1 k=1 k=1
1 n
A = {n : n " N} F = {n+k : n, k " N} K = {n-2k : n, k " N}
n+k
n
k

B = {n+1 : n " N}
G = {n : n, k " N, k < n}
2n+3k
L = : k, n " N
4n+k
1 1 1
C = {1 + : n " N} H = {n + : n, k " N}
k
n2
2 1 1
M = (-1, 2) )" Q
I = {n - : n, k " N}
D = {n +2n-3 : n " N}
k
n+1

4n-3k
nk
J = : k, n " N
E = {1+n+k : n, k " N} N = (-1, 1) )" (R \ Q)
2n+5k
"
4n - 3k
3
: k, n " N = - , 2 )" Q.
2n + 5k 5
n n
(-1)n
A = {2 : n " N} D = {1 + : n " N} G = {2n2 : n, k " N}
n!
n2 +2k
B = {21 : n " N} E = {(-1)nn : n " N}
n
1 1 1 n!k!
C = {n + + : n " N} F = {1+n!+k! : n, k " N}
2n 3n
 a " Z a " N -a " N a = 0
Z )" R+ = N Z )" R- = -N
a, b " Z a + b " Z ab " Z a - b " Z
1
" Z
/
2
2N = {2n : n " N} 2N - 1 = {2n - 1 : n " N}
n, m " N nm " 2N n " 2N m " 2N
x " R n " N q, r " N p " Z
r n

x p < 1

0 < -
,

q qr
x
x " R n " N
p " Z q " N q n

x - p < 1

.
q qn
"
x " R n " N
p " Z, q " N q n

x - p < 1

.

q q2
(0, 1) (-1, 3) (a, b) (c, d)
"
(0, 1 (0, 1)
R sup " inf " sup " inf "
x " R x = 0 a " Z

1
xa = , x � xa = x1+a.
x-a
x " R x = 0 m, n " N

xm
xm-n = .
xn
x, y " R x, y = 0 a, b " Z

xa � xb = xa+b xa � ya = (x � y)a (xa)b = xab
xa 1 xa
= xa-b = = (x)a
ya y
xb xb-a
 x > 0 n " N x > 1 xn > 1
x > 1 a, b " Z a < b xa < xb

" " "
" "
n
n m nm
n n
x � y = x � y x 0 y 0 x = x x > 0

n 1 1
"
= x > 0
n
x
x
"
"
nk
n
n " 2N - 1 k " 2N x < 0 x = xk

"
2 " R \ Q
"
1
x = x x " R xr � yr = (x � y)r r " Q x, y > 0
"
|x| = x2 x " R
xr
= (x)r r " Q x, y > 0

yr y
"
x n " 2N - 1
n
xn =
|x| n " 2N, (xr)s = xr�s r, s " Q x > 0
xr � xs = xr+s r, s " Q x > 0
r < s xr < xs
xr 1
= xr-s = r, s " Q x > 0 r, s " Q x > 1
xs xs-r
a > 1 b " R
n " N an > b
x > 1 y " R r " Q
r < y �! xr < xy y < r �! xy < xr
y, z " R 0 < x < 1 y < z xy > xz
x > 1 y " R
xy = inf{xr : r " Q, r > y}.
x, y " R \ Q xy " Q
"
xy = yx
1 n n+1 x, y x < y
1
x = 1 + y = 1 +
n n
x < y
x > 0 y 2y = x
loga(x � y) = loga x + loga y a > 0 a = 1 x, y > 0

loga(xy) = y loga x a > 0 a = 1 x > 0

1
loga b = a > 0 a = 1 b > 0 b = 1

logb a
logb x
loga x = a > 0 a = 1 b > 0 b = 1 x > 0

logb a
 a > 1 0 < x < y loga x < loga y
0 < a < 1 0 < x < y loga x > loga y
f g f ć% g = g ć% f f ć% g = g ć% f

"
f(x) = [ x] x " 1, 10 [a]
a
f(x) = x - [x] x " R

1 x " Q
f(x) =
0 x " R \ Q
y = f(x)
y = f(-x) y = -f(x) y = f(x/2) y = f(2x) y = f(|x|) y = |f(x)|
f(x) = xą x > 0
ą > 0 ą < 0 ą = 0
x4 + 1
f(x) = a0 + a1x + � � � + anxn x " R an = 0

x0 " R f
n-k
ak 1
|x0| 2 max : k = 0, . . . , n - 1 .
an
a " R f
f(a) = 0
"
"
"
xy R3
x-3
f(x) = x " (1, 3 f(x) = x2 + 4x + 1, x " -6, -4)
x+1
1
f(x) = x3 - 3x2 + 3x + 27, x " R
f(x) = x + , x " 1, +")
x
"
x
f(x) = x " R f(x) = - x + 1, x " 0, +")
1+|x|
"
f(x) = 2 - x - 4, x " 4, +")
f(x) = x|x| x " R
"
3
f(x) = x x " R
f(x) = x2 + 2x - 3 x " (-5, -2
1
f(x) = -x2 + x + 6, x " -3, -1) f(x) = (1 - 5x), x " (-", 0
2
1
f(x) = -x2 + x + 6, x " -3, 1) f(x) = , x " (-", 1)
2x+4
1
f(x) = log3(4 - x2), x " (-2, 2)
f(x) = 4x-1 , x " 0, 1)
x+1
1
f(x) = 1 + , x " 3, 9)
f(x) = 2x-1 x = 1

log3 x
x
f(x) = log2 x2 x = 0 f(x) = log , x " (0, +")

x+1

f(x) = x - [x], x " R f(x) = [x] + x - [x], x " R

f(x) = x - [x], x " R f(x) = |x| - [x], x " R
21.11.2011 - 25.11.2011.
(an) an+1 an
an+1 an n " N
n 2n+1 1 1 1
an = log2 n+1 an = an = + + � � � + +
n+2 1! 2! n!
2

(n+1)!
2n
n
an =
an =
3n
n+1
" n!(2n)!
n
an =
an = 5 (3n)!
"
"
a1 = 2 an+1 = 2 + an
"
n
an = (-1)n 1-n
an = 1 + 6n
n2
1 1 1
an = n2 - n + 1 an = + + � � � +
n+1 n+2 2n
n
(-1)n
4n+1 2n
1
an = bn = dn =
en = 1 +
n+1 n 3n
n
n+1
log2(n)-1 1
n-1 1
fn = 1 +
an = cn = n + gn =
n
n+2 n log2(n)+5
"
a1 = 1 a2 = 1 an =
an-1 + an-2 n = 3, 4, . . .
"
1 "
1 + 5 n 1 - 5 n
an = " - .
2 2
5

4-n2
1
limn" 2n-5 = 2 limn" n2+n = -1
limn" n = 0
n+1
limn" n3-2n2+4 = 0
limn" -n2+3n-2 = -1
limn" log2(n)-1 = 1
4+2n2 2 n4-5n2+1
log2(n)+5
3
limn" 3n+5 = limn" 3n4-2n3+n-8 = 3 limn" 3n+5 = -1

2n-1 2 n4-2 2n-1
limn" an = g an A n g A.
limn" an = g, limn" bn = g an bn n g g .
"
"
an 0 limn" an = a limn" an = a
"

2n3+3n2-5
limn" (n-1)(n2+2) limn" 3 n3 + 5 - n limn" 3n-2n
4n-3n
limn" 1+2+���+n
limn" (n+1)!-n! n2
(n+1)!+n!
" limn" log4(n+1)
log5(n+1)


" "
limn" 1+2+���+n
limn" n + n - n n
" " "
"
3 22 2n
limn" 1+3+5���+(2n-1)
limn" 2 2 2 � � � 2
2+4+6+���+2n
limn" n2(1+(-1)n)
n+1
1 1 1

1+ + +���+
1 3n
1
limn" 1+ 2 22 2n
limn" 2n(-1)n - 1 1 1
limn" n log2 n
+ +���+
6n+1
3
32 3n
(an) an > 0 n " N
limn" an+1 = g
an
1
g < 1 limn" an = 0 g > 1 limn" an = 0
n2 (n!)2
limn" (2n)! limn" 10n
limn" (2n)!
n!
(an) (bn) an - bn 0
an an
0 +"
bn bn
an an
2
bn bn
an
(an) (bn) an + bn
bn
(n!)2
an =
(2n)!
limn" an = g limn" an+3 = g
"
"
an
a1 = 2 an+1 = 2 + an n " N a1 = 2 an+1 = n " N
1+an
"
1 1
"
a1 = 2 an+1 = (an + ) n " N
3
2 an
a1 = 3 an+1 = an + 6 n " N
a1 = x0 an+1 = an(2 - an) n " N
3+an
a1 = 2 an+1 = n " N x0 " (0, 1)
4
a > b > 0. a1 := a, b1 :=
"
an+bn
b, an+1 := , bn+1 := anbn limn" an = limn" bn.
2
E R
M E
x M x " E.
(an) an " E, limn" an = M.
"
(an)
limn" an = g limn" a1+a2+...+an = g.
n
limn"(an+1 - an) = g limn" an = g.
n
"
(an) an > 0 n " N
"
limn" an = g limn" n a1a2 � � � an = g.
"
limn" an+1 = g limn" n an = g.
an
1
+ (-1)n
n
2
(an) limn" a2n = g limn" a2n+1 = g
limn" an = g
(an) (a2n), (a3n), (a2n+1)
(an)
(an) (asn) s > 1
(an)
(an)
(an)
(an) an = n n " N mk = (2 + (-1)k)k lk = k + (-1)k+1
(amk)k"N (alk)k"N (an)n"N
(-1)n
1
(n+(-1)n)(-1)n
1+
1
n
an =
an = 1 +
an =
n
n
2+(-1)n
"
[ 2n] ((-1)nn+1)(-1)n
an = an =
n n

limn" 3n+(-1)n
limn" n (1)n + (3)n
5n+1
2 4

"
1 1 1
limn" n 4n2 + n + 5
" "
limn" "n2+1 + + � � � +
n2+2 n2+n
"
limn" n 5n - 3 � 2n + 3n

1 1 1
" limn" (n+1)! + + � � � +
(n+2)1 (2n)!
limn" n 2 � 5n - 3 � 2n
"
n5
limn" n 7n + 5n - 3n+2 limn" 2n+3n


1
limn" n 1 + 2n +
3 2n-1
2n
limn" 1 � � � �
2 4 2n

2 3 n
limn" n 1 + + + � � � +
1
2 3 4 n+1
limn" n[nx] x " R
"
1
limn" n2 n2 + 1
[n2+ ]
limn" 2n2n [x]
limn" log2(2n+1) x
log2(4n+1)
limn" n2 = +" limn"(5 - 2n) = -"
1
limn" log2 n = -"
limn" 2n2+3 = +"
3n-1
limn" log 1 n = -"
limn" n3+2n2+4 = +"
2
n2+3n+5
"
limn" n-n3 = -"
limn" 7n2 - n + 3 = +"
2n+1
n2+1
limn" 7n5-13n4+5n-2 = +" limn" 1-3n = -"
n3+2n-5
an, bn +"
an an
an - bn 2 an - bn +" 2 0
bn bn
(an)
(an)
(an)
(an) g (mk)k"N
+" limk" amk = g.
limn" an = +" (bn)
limn"(an + bn) = +".
limn" an = +" an bn n
limn" bn = +".
limn" an = a a > 0 limn" bn = +" limn" anbn = +"
limn" an = -" limn" bn = +" limn" anbn = -"
"
"
[n3 2]
limn"(n - n)
limn" ["n2+1] [x]
x
limn"(n4 + (-1)n � n)
limn"(7n - 6n - 5n)
"
limn" n n!
1 1 1
" " "
limn"(1 + + + � � � + )
n
2 3
"
limn" (2n+1)3n limn" n2 n!
n(2n+1)
(an)
� > 0 n0 m > n0
|am - an0| < �.
(an) (an+1 - an) 0 n ".
(an)
an+1 - an 0
1 1 1
an = 1 + + + � � � + .
22 32 n2
e
2n+1 6n
2n2
+5
1
limn" 3n-4 limn" 1 +
limn" n2+3
3n+1 2n+3
n2+1
n
n
limn" 5n-(-1)n�7
n2
5n+2
4n
n2+2
limn" 4n+1
limn" 2n2+1
4n
limn" n-5
n+3
-n+3
2+4n
(n+1)2n
4
limn" (n2+2n)n2+2n
limn" 1 -
limn" 3nn!
n
nn
n " N
n n n n
< n! < e � .
e 2
lim supn" an = inf{x " R : n an x}.
lim infn" an = sup{x " R : n an x}.
(an)n"N
lim supn" an = inf{sup{ak : k n} : n " N}
lim infn" an = sup{inf{ak : k n} : n " N}

n(n-1)
n
2 an = 1 + 2n(-1)n
an = 1 + 2 � (-1)n+1 + 3 � (-1)
n
" "
(-1)n
an = 1 + an = n - [ n]
n

n n
6" an = nr - [nr] r " Q
an = -
3 3

7" an = nr - [nr] r " R \ Q
2n2 2n2
an = -
7 7
 x0 E
x0 E
E �" R
E
(an)n"N X = {an : n " N}
E X E
(an)n"N
n n
E = {(-1) : n " N} E = {1-(-1) : n " N}
n 2
2n+1
E = {(-1)n + : n " N}
n E = {x " R : 1 < x 2}
n
E = {2n : n " N}
1
E = {1 + : n " N}
n2
n
E = {(-2) +n : n " N}
2n
E = {3n-2 : n " N}
n+2
1
E = {(-1)n(1 + ) : n " N}
n
3m
E = {2n+4m : n, m " N}
n
E = {4+(-1) n : n " N}
n+1

3m-2k
E = : m, k " N
E = {3n+2 : n " N}
4m+k
5n+1
E I
E
3n+1
E = {(-1)n + : n " N \ {1}}, I = (4, +")
n-1
5n+1
E = {(-1)n + : n " N}, I = (6, +")
n
3n+1
E = {(-1)n + : n " N}, I = (2, 4)
n
" "
2 E = {(- 2, 2) )" Q}
"
1 E = {(- 2, 2) )" (R \ Q)}
1
x0 " {0, 1, , 3} (0, 1)
2
[0, 1], (0, 1) Q
" "

1 1
ln(1 + ) i " .
n n
n=1 n=1
1 1 1
(1 + )n < e ln(1 + ) < n " N
n n n
1
ln(1+ )
limn" 1 n = 1.
n
" 1
" 1 A n=1 2n
B C. A = B + C
n=1
2n-1
2B = A, B = C.
" 1 " 3n+2n
n=1 2n n=1 6n
" 10n+1
"
n=1 n
(-1)n
n=1
" 1
n=1 "
n(n+1) " "
( n + 1 - n)
" 1 n=1
n=1
n(n+1)(n+2)
"
1
"
" "
" n2+n+1
n=1
n(n+1)( n+ n+1)
n=1
n(n+1)
" " "
" "
n
( n + 2 - 2 n + 1 + n)
n=1
(2n-1)2(2n+1)2 n=1
1, 3(27)
" n2+1 " ln n "
1
" "
n=1 n=1 n=1
n3
n2-n+ 2
(2n-1)(2n+1)
" nn
" "n+1-"n
" 1
n=1
3nn!
n=1
n
n=1
n2-4n+5
" (2n)!
" 1+n
n=1
n2n
n=1 " "n
1+n2
" (2n)!
n=1
n2+1
" 1
n=1
(n!)2
n=1 " 1
n4+1
" n2
"
n=1 n
2
" 1 n+1
1
n=1
(2+ )n
n
(n - ln )
n=1 n
" 1
n-1 n2
"
"
n=2
(ln n)ln n
1
e2n
ln2 (1 + )
n=1
n
n=1
n
" 1
" n-1 " n5
n=2
n(ln n)p
n=1 n=1
2n+3n
n3+2
"
" 1
1 " 1
"
"
1 n=1
a, b, s > 0
n=1
n( n2+n n-n) n=1 a+bns
n1+ n
1 1 1
< ln(1 + ) < .
n+1 n n
" n
2(-1) -n
n=1
n
" n " 3n " 1�3���(2n-1) " ln n
"
"
( n - 1)n, , , 1 - .
n=1 n=1 n=1 n=1
n2n 3nn! n
" 2n
" 1
" (n!)2
,
,
n=1 ,
nn
n=1
1+xn
n=1
(2n)!
x > 0,
" n-1 n(n-1)
" (n+1)!
,
,
n=1
n+1
n=1
2nn!
"
"
"
n " (-1)n+12n2
(-1)n( 3 - 1),
(-1)n (1-n2),
n=1 ,
n=1
n2
n=1
n!
"
" (-1)n+1
"
(-1)n ln n,
,
n=1
n
n=1 n (-1)n (n+1)n
n=1
nn+1
"
" (-1)n+1
" ann!
(-1)n 2+(-1)n ,
,
n=1
n=1 n
n(n+1)
, a " R,
n=1
nn
"
" (-1)n+1
" ną
(-1)n 2+(-1)n ,
,

n=1 , ą = 0,
n2
n=1
n ln(n+1)
n=1 ąn
"
" (-1)n
"
(-1)n 1+n, "
16" (-1)[log2 n] 1
n=1
n2 n=1
n-(-1)n n n=1 n
" (-1)n n n2
" (-1)n
" (-1)["n]
"
,
,
17"
n=2
3n n-1 n=2
n+(-1)n n=1
n
n n
n n
< n! < e
e 2
" " an
a2 .
n=1 n n=1
n
"
" n=1 an
a2
n
n=1
"
" (an)n"N n=1 an
a2
n
n=1
"
" an n=1 an
n=1
1+an
"
" n=1 an
ln(1 + an)
n=1
" 1 (an)
n=1
nan
" 1
"
n
n=1
n n
"
an Sn
n=1
" an
n=1
Sn
"
" n=1 an (bn)
anbn
n=1
"
"
" n=1 an
ćn
n=1
�
S.
" (-1)n
"
n=0
n+1
"
1
| q |< 1 nqn-1 = .
(1-q)2
" n=1
qn.
n=0
x " R
" "

xk (-x)k
� = 1.
k! k!
k=0 k=0
 " (-1)n " (-1)n ą + � < 1, ą > 0, � > 0
n=0 n=0
(n+1)ą (n+1)�
" "
an bn
n=1 n=1

1 n = 1
an = 3 n-1
- n = 1

2
n-2
3 1
bn = 2n-1 + .
2 2n
" "
an bn
n=1 n=1
" xn " n2
" (n!)2
xn
xn,
n=1 n=1
n! (n+1)22n
n=1
(2n)!
"
" (-1)n
n!xn
n=1
(n)nxn,
n=1
n! e
" 1
" x2 n
(x - 2)n
n=1
n2n+1
" xn2
n=1
n
,
n=1 2n
" xn
" xn
, p " R " 2n
n=1 ,
np
n=1 2n+3n
xn
n=1 1+3n
n+1 n2
" 1
" n!xn
xn
n=1
3n n
n=1
nn
" 1 2 " 1 " n2+2
(1 + )n xn, xn (x + 1)n
n=1 n=1 n=1
n n10n-1 n+1
"
�
e
n! = 2Ąn(n)ne12n , 0 < � < 1,
n!
limn" "2Ąnnne-n = 1 (n)n < n! < e(n)n.
e 2
1 n
" 1 "
(2x + 10)n (1 - x)n
n=1 n=0
n2 3
" "
(6 - 3x)n x2n
n=1 n=1
q
2n
1
"
1 3 sin
1-cos
n+1
limn" 1 + sin
n 7. limn" sin "1
limn" sin 1 n
2n
n
cos nĄ
"
1
5
5" limn" 1 +
n
limn" sin2 n+4n 8. limn" n cos(n+1)
3n-1 n+1
"
3n+1
limn" (5+cos n)n 6" limn" cos(Ą n) 9. limn" sin n
n
sin 1 sin 2 sin n
an = + + � � � + .
2 22 2n
" n+2 " " 1 1
1
cos n2 sin sin cos
n=1 n=1 n=1
n2+1 2n n n

" " x " 1
1
cos sin sin , x " (0, 1) tg2
n=1 n=1 n=1
n n n
 " 1 " cos n
" "2n+1 n
"
tg2
sin
n=1 n=1
n!
n
n=1
4n+1
" 1 " 2+cos n!
" sin2 n
"
sin
n=1 n=1
n2
n
n=1
n

" (sin 1)n " " sin n
1
2
1 - cos
n=1 n=1 n=1
2n n n
n n
n+1 n n+1 n

sin x sin x cos x sin x
x
2 2 2 2
sin kx = , cos kx = , sin = 0.

x x
sin sin 2
2 2
k=1 k=1
" "
an cos an
n=1 n=1
"
" " " n=1 an
sin an sin 3an tg an
n=1 n=1 n=1
"
" n=1 an
sin(nan)
n=1
"
f(x) := limn"(limk"(cos n!Ąx)2k), x " R,
"
(sin n)n"N
(sin nx)n"N x = kĄ k " Z

f : R R
limx0 f(x) = 3 limx-1 f(x) = 2 limx3 f(x) = -1
2
1
limx2 3x = 6 limx4 x-3 = limx2 x2-4 = 4
x+2 6 x-2
limx3 x2 = 9 limx0 x5+3x+1 = -1
limx-1 x4-3x2+8 = -2
x-1
x2-4
3 3
limx4 x = limx1 x2+5 = 2 limx-2 x+3 = -1
4 x+2 x-5 7
f, g : E R x0 E.
f(x) g(x)
x " E \ {x0} limxx0 f(x) = a limxx0 g(x) = b a b
f : E R x0 E. limxx0 f(x) = g
 > g (a, b) x0 " (a, b) f(x) <  x " (a, b) )" E x = x0

f : E R, x0 " E E
limxx0 |f(x)| = 0, limxx0 f(x) = 0.
 f : R R
limx3 f(x) = +" limx1- f(x) = -" limx-2 f(x) = +"
1
x3-x2+x-2
limx2- = +"
limx0 ex2 = +"
x2-4x+4
x+5
limx2+ = +"
x-2
1
"
limx0+ = +"
x+1-1
x+5
limx2- = -"
x-2

1
x3-x2+x-2
limx0 ln 1 + = +"
limx2+ = +"
x2
x2-4x+4
f : R R
3
limx-" f(x) = 1 limx+" f(x) = -3 limx+" f(x) =
4
+" -"
5x2-3 5
limx+" 4x2-4x+1 = 2 limx+" 3x2+x =
2x2-x 3
1
limx-" 4x2-4x+1 = 2 limx-" cos = 1
2x2-x x
+" -"
f : (a, b) R
limxb- f(x).
f : (a, +") R a " R f
f +".
f : R R
limx-" f(x) = +" limx+" f(x) = -"
limx-" f(x) = -" limx+" f(x) = +"
+" -"
"
limx+" x4+x = +" limx-" 1-x6 = -" limx-" 1 - x = +"
x-1 x4-1
+" -"
 limx0 sin x limxa sin x a " R limx+" sin x
x
limx0 sin 3x limx+" ln(1+ex)
x
2x
limx0 tg 5x
limx+" ln(2+e3x)
ln(3+e2x)
limx Ą cos x-sin x
cos 2x
4
limx0 (x+1)n-1 n " N
x

1 1
limx Ą - +
cos x ctg x
limx1 x+x2+���+xn-n n " N
2
x-1
limx Ą cos x
Ą
x-
2
2 limx0 (1+x)(1+2x)���(1+nx)-1 n " N
x
"

limx0 cos x-1
1 1
x2
limx0 x -
x
"

limx0 1-cos x
sin x
1
limx0 x
x
sin2 x
limx0 1-cos x
limx+" [3x]
[x]
x2+sin
limx+" x2-cos x
x
1
limx0(1 + sin x)x
limx0 sin x
|5x|
1
limx0(cos x)x
x
limx0 sin 5x-sin 7x
1
"
" "
limx0(cos x)x2
limx+" x � sin( x + 1 - x)

"
"
3
"x-10
limx+" x(x - x2 - 1)
limx106
3
x-102
"
"
3
limx-"(x + x2 - 3x)
limx0 1+mx-1 m " R
x
" "
" "
limx+"( x + 3 - x + 1)
limx2 x-1- x2-3
x-2
"
"
limx+"(x - x2 + x)
1 n
limx0 x 1 - 1 + x
"
limx-"(x + x2 + x)
limx0 25x-9x
5x-3x
"
4
limx-" x4+1
x
limx0 e6x-1
ex-1
1
1 limx+" x sin
x
limx0+ ex
1
limx0 x sin
1
x
limx1- e1-x2
1
limx0 sin
1
x
limx1+ e1-x2
x
limx1 cos
x-1
limx0 ln(1+x)
x
limx+" cos x2
limx0 ex-1
x
limxĄ 2ctg x
limx2 2x-x2
x-2
x+1

limx+" 3x-4 3
3x+2
limx0 1-e2x
tg x
2x-1
limx+" x+1
limx10 log x-1
x-2
x-10
f x0
1
"
f(x) = x2 - 2x x0 = 0 f(x) = x0 = 4
x
x
x2-x
f(x) = x0 = 2
f(x) = x0 = 3
x-1
x-2
1 2x3-x2-3x+6
f(x) = x0 = 1 f(x) = x0 = 1
x x3+1
|�| : R R
f : E R x0 " E, P
x0 f|(E )" P )
f : E R x0 " E f(x0) > 0
P x0 f|(E )" P )
x0

"
1
f(x) = 2 - x x0 = 1
sin x = 0

x
f(x) = , x0 = 0,
0 x = 0

x2-4 1
, x = -2 x sin x = 0

x+2 x
f(x) = , x0 = -2, f(x) = , x0 = 0.
-4 x = -2 0 x = 0
f, g : E R x0 " E
max(f, g) max(f, g)(x) = max(f(x), g(x)) x0.
min(f, g).
max(f1, . . . , fn)
min(f1, . . . , fn).
"
max min sup inf
x " R

3x x " Q f(x) = sgn(sin x) x " R
f(x) =
0 x " R \ Q,
f(x) = [x] sin Ąx x " R

x2 0 x 1
f(x) =
x2
2 - x2 1 < x 2, f(x) = limn" 1+xn x " (-1, +")
f : R R
x0 = 1 x1 = 1 x2 = 2 xk = k k " N
 f : R R
k k " N
f(x) = [x], x " R R
Z
a, b, c f : R R
ńł
Ąx
�łsin ax
cos |x| 1 x < 0
�ł
2 x
�ł
f(x) =
�ł
�ł x3-1
a|x - 1| |x| > 1, 0 x < 1
x2+x-2
f(x) =
�łc
�ł
�łx2+(b-1)x-b x = 1
�ł
ół
ńł x > 1,
1
x-1
�ł2 + ex x < 0
�ł

sin ax x3+8
f(x) = x = 2
x > 0
3x x-2
�ł
f(x) =
ół
b x = 0, a x = 2,
f : R R f(x+
y) = f(x) + f(y) x, y " R f(x) = ax, a " R
"
f : [0, 1] R
ńł
0, x
�ł
�ł
�ł
f(x) =
p
1
x = , p " Z, q " N
�ł ,
q
�ł q
ół
p
q
[0, 1]
f : A R A
x3
f(x) = x3 A = [1, 5] f(x) = A = R
x2+1
f(x) = x3 A = (-", 0)
x4
f(x) = A = [1, 5]
x2+1
f(x) = x3 A = (0, +")
x
f(x) = A = R
x2+1
1
f(x) = A = [�, +") � > 0
x
x2
1
f(x) = A = [-2, 5]
f(x) = A = (0, �) � > 0
x2+1
x
x2-1
f(x) = A = R f(x) = sin x A = R
x2+1
"
f(x) = x x " [0, 1]
f : A R A �" R
A f
 f : A R A
Ą 1
f(x) = sin A = (0, 1) f(x) = A = [a, +") a > 0
x x
f(x) = x sin x A = [0, +")
f(x) = x2 A = R
f(x) = x + sin x A = R |x|
f(x) = A = R
|x|+2

Ą
f(x) = tg x A = -Ą ,
2 2
f(x) = sin x A = R
1
f(x) = A = (0, 1]
x
f(x) = 3x + cos 5x A = R
4
f(x) = A = (1, 2)
2-x
f(x) = sin(sin x) A = R
f(x) = ax A = [0, +") a " (0, 1)
f(x) = sin(x sin x) A = R
f(x) = 2x A = [0, +")
"
f(x) = sin x2 A = R
f(x) = x A = (0, 1)
"
1
3
f(x) = x sin A = [1, +") A = (0, 1]
f(x) = x A = [2, +")
x
f(x) = ln x A = (0, 1) A = [1, +") 21" f(x) = x ln x A = (0, 1]

1
sin , x " (0, 1],
x
f(x) =
0, x = 0
[0, 1],
f : P R P
m = infx"P f(x), M = supx"P f(x)
(m, M) �" f(P )
"
f : [a, b] R
ln x + cos Ąx + 2 = 0
f : [a, b] [a, b] a < b
x0 " [a, b] f(x0) = x0
"
f : R R f(f(x)) = -x x " R
f
f : R R
f : R R
f : R [0, +") R
[0, +") f
 x
" 1
f(x) = x g(x) =
16
"
x
1
f(x) = log 1 x g(x) =
16
16
1 1
f g x = x =
2 4
f x0

1
f(x) = x - [x] x0 " Z
cos x = 0

x
f(x) = x0 = 0,

x
0 x = 0
e1-x x = 1

f(x) = x0 = 1,
0 x = 1

1
(x2 - 3) sin Ą x = 0

2x
sin x
f(x) = x0 = 0
x = 0

x
0 x = 0
f(x) = x0 = 0,
0 x = 0

1
1
ln x = -1

arctg x = 0

(x+1)2
2x
f(x) = x0 = -1, f(x) = x0 = 0.
Ą
1 x = -1 x = 0
4
f(x) = arcsin(x + 1) f(x) = sin(arcsin x) f(x) = [sin x]
f(x) = | arcsin x| f(x) = arcsin(sin x)
1 1 Ą
4 arctg - arctg =
5 239 4
Ą
arcsin x + arcsin 2x =
2
x
arcsin x + arccos x = Ą/2, x " [-1, 1] arcsin x = arctg("1-x2 ), x " (-1, 1)
ex-1
limx0 arccos limx0 arcsin 3x
2x x

1
1
limxĄ x - arcctg
limx-1(1 + x) arctg
x-Ą
1-x2
02.01.2012 - 06.01.2012.
16.01.2012 - 20.01.2012.