lim infn" fn(x) a" 0, lim infn" fn(x)l(dx) = 0.
R
ńł

1
�ł 1 l(dx) = , n-parzyste
1
�ł [0, ] 2
2
fn(x)l(dx) =

�ł
1
R
ół
1 l(dx) = , n-nieparzyste.
1
[ ,1] 2
2

1 1
fn(x)l(dx) = lim infn" R fn(x)dx = .
R 2 2

lim inf fn(x)l(dx) d" lim inf fn(x)l(dx)
n" n"
x " R fn(x)
limn" fn(x) = 0.

lim fn(x)l(dx) = 0.
n"
R

1
fn(x)l(dx) = n � = 1,
R n

lim fn(x)dx = 1.
n"
R

limn" f(x)l(dx) = limn" f(x)l(dx).
x > 0
(1 + x)n (1 + x)n+1
d"
1 + (1 + x)n 1 + (1 + x)n+1
(1 + x)n + (1 + x)2n+1 d" (1 + x)n+1 + (1 + x)2n+1
(1 + x)n d" (1 + x)n+1.

(1 + x)n (1 + x)n
lim e-xl(dx) = lim e-xl(dx)
n" n"
1 + (1 + x)n 1 + (1 + x)n
(0,+")
(0,+")
= e-xl(dx) = 1.
(0,+")
 fn(x) = e-|x|(sin x)n.
|fn(x)| d" e-|x|. fn(x) -n" 0, l - p.w.
Ą
x = + 2kĄ sin x = 1, fn(x) -n" e-|x|
2
x = -Ą + 2kĄ sin x = -1, {fn(x)}
2

lim e-|x|(sin x)nl(dx) = lim e-|x|(sin x)nl(dx) = 0.
n" n"
R R

m

1
I[ 1 1 (x)
2n+1 2n
4n , ]
n=1
m

" m
" "

1 1
I[ 1 1 (x)l(dx) = lim I[ 1 1 (x)l(dx)
2n+1 2n m" 2n+1 2n
4n , ] 4n , ]
0 0
n=1 n=1

m m
" "

1 1
= lim I[ 1 1 (x)l(dx) = lim I[ 1 1 (x)l(dx)
m" 2n+1 2n m" 2n+1 2n
4n , ] 4n , ]
0 0
n=1 n=1

" "

1 1 1 1 1 1
= l , = � = .
22n 2n+1 2n 22n 2n+1 14
n=1 n=1

" n n

cosk x l(dx) = lim cosk x l(dx) = lim cosk x l(dx)
Ą 3 n" Ą 3 n" Ą 3
[ , Ą] [ , Ą] [ , Ą]
k=0 4 4 k=0 4 4 4 4 k=0

n "

= lim cosk x l(dx) = cosk x l(dx),
Ą 3 n" Ą 3
[ , Ą] [ , Ą]
4 4 k=0 4 4 k=0
"
2 Ą
cos x " [0, ], x " [Ą , ]
2 4 2
cos x < 0

"

l(dx) l(dx) 1 1
cosk x l(dx) = = = - .
Ą 3Ą
Ą 3 Ą 3 - cos x 2 sin2 x tan tan
Ą 3
1
[ , Ą] [ , Ą] [ , Ą]
2 8 8
4 4 k=0 4 4 4 4
"
� = �k
k=1

"


x e-1
exp -x 2 + cos d" e-x i e-x�(dx) = e-k = ,
n 1 - e-1
[1,")
k=1

"

k x
lim exp {-k(2 + cos )} = lim exp {-x(2 + cos )}�(dx)
n" n"
n n
[1,")
k=1

x
= lim exp {-x(2 + cos )}�(dx)
n"
n
[1,")

"

e-3
= e-3x�(dx) = e-3k = .
1 - e-3
[1,")
k=1
"
� = �k
k=1


"

1 min(k, n) 1 min(x, n)
lim 1 - = lim 1 - �(dx)
n" n"
k2 n x2 n
[1,"]
k=1


1 min(x, n)
= lim 1 - �(dx)
n"
x2 n
[1,"]

"

1 1 Ą2
= �(dx) = = .
x2 k2 6
[1,"]
k=1