шамин с сдо

|f(x) − A| < ε.

−∞

f(x)

(−∞, a)

A

x → −∞

ε > 0

M > 0

x < −M

|f(x) − A| < ε.

arctg(x)

lim arctg(x) = π/2,

x→+∞

x lim arctg(x) = −π/2.

→−∞

x→0

x)

1

z

= z→∞ 1 + z

lim (1 +

1/x

lim

.

e

1

z

z→+∞ 1 + z

.

z > 0

lim

n N

n ≤ z ≤ n + 1

1

n

1

z

1

n+1

1 +

≤ 1 +

≤ 1 +

.

n + 1

z

n

1.5

1.0

0.5

x

arctan

0.0

−0.5

−1.0

−1.5

−100

−75

−50

−25

0

25

50

75

100

x

un =

1 + n1 n

z

·

1

≤

1

1

≤

1 + n+11

z

n

un+1

1 +

un

1 +

.

z → +∞

n → ∞

nlim un = e

→∞

1

z

z→+∞ 1 + z

= e.

lim

1

z

z→−∞ 1 + z

.

lim

z = −(t + 1)

t → +∞

z → −∞

z→−∞

1

z

1

−(t+1)

z

= t→+∞ 1 − t + 1

=

lim

1 +

lim

t

−(t+1)

t→+∞

1

t+1

= t→+∞ t + 1

t

lim

t

= lim

1 +

=

t→+∞

1

t→+∞

1

t

·

t

= lim

1 +

lim

1 +

= e.

x0

±∞

f(x)

x → x0 x0

±∞

xn

f(x)

xn → x0

nlim f(xn) = ∞.

→∞

n

lim

x

= lim xn−m = 0.

m

x→0 x

x→0

1.0

0.8

0.6

sin

0.4

0.2

0.0

−0.2

−20

−15

−10

−5

0

5

10

15

20

x

f(x)

x0

f(x)

(a, b)

(a, b)

x0 (a, b)

f(x)

[a, b]

[a, b]

(a, b)

a

b

f(x)

[a, b]

|f(xM )| > M.

M = 1, 2, . . .

{xn}

|f(xn)| > n.

{xnk }

klim xnk = x0

x0 [a, b]

→∞

f(x)

x0

lim f(xnk ) = f(x0).

k→∞