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Fn : [0, +∞[→ R Fn(ρ) = 2n ρ
1+n2 n ρ 2 Fn(0) = 0 lim
ρ→+∞ Fn(ρ) = 0
F ′ n(ρ) = 2n − 4nnρ2 (2nnρ2 2 ,
+ 1)
ρn = 1/ √ n2 n
Fn
sup
ρ≥0
θ∈[0,2π]
|fn(ρ cos θ, ρ sin θ)| = √ 2Fn (ρn) = √ 2
2 n
2 √ .
n2n +∞ n → +∞ R 2
{(ρn cos θ1, ρn sin θ1), (ρn cos θ2, ρn sin θ2)}
(0, 0)
(0, 0)
(0, 0) ¯ρ > 0
sup |fn(x, y) − f(x, y)| = sup |fn(x, y)| = sup |fn(ρ cos θ, ρ sin θ)|
(x,y)∈R\B((0,0),¯ρ)
(x,y)∈R\B((0,0),¯ρ)
ρ≥¯ρ
θ∈[0,2π]
= √ 2 sup
ρ≥¯ρ
2 n√ 2ρ
1 + n2 n ρ 2 = √ 2 sup
ρ≥¯ρ
Fn [ρn, +∞[ ρn
n ρn < ¯ρ Fn
[¯ρ, +∞[ Fn(¯ρ) ≥ Fn(ρ) ρ ≥ ¯ρ
F (ρ)
sup |fn(x, y) − f(x, y)| =
(x,y)∈R\B((0,0),¯ρ)
√ 2
2 sup
ρ≥¯ρ
n√2ρ 1 + n2nρ2 = √ 2F (¯ρ) = √ 2
2 n ¯ρ
1 + n2 n ¯ρ 2
|fn(¯ρ cos θ1, ¯ρ sin θ1)|,
R
fn(x) = nxe −n2x2 fn : R → R
fn(x) =
nx
1 + n 2 x 2 fn : R → R
fn(x) = nx
1 + nx fn : [0, 1] → R
fn(x) = (x 2 − x) n fn : [0, 1] → R
x ∈ R |fn(x)| n → ∞
f(x) = 0 R fn