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cosh x = cos(ix)
x = −i arcsin(ix)
x = −i arctan(ix)
x = 1
2
fn(x) = n
= n
=
x + 1
n − √
x
x + 1
n − √ x
1
x + 1
n + √ x
f(x) = 1
2 √ x
=
log 1+x
1−x
x + 1
n + √
x
=
x + 1
n + √ x
n→∞
−→ 1
2 √ x .
∀ x > 0
fn f E ⊆ (0, +∞) ⇔ lim
|fn(x) − f(x)| =
1
x + 1
n + √ x
− 1
2 √
=
x
n→∞ sup
E
1
2n √
x x + 1
n + √ 2
x
|fn(x)−f(x)| = 0 .
√
1
1 x+
−→ 0
(0, +∞) sup (0,+∞)
2n √ x
n +√x
[a, +∞) a > 0
sup
[a,+∞)
1
2n √
x x + 1
n + √ x
n≥1 un(x) :
2
=
1
2 = +∞ n→+∞
2n √
a a + 1
n + √ 2
a
n→+∞
−→ 0 .