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α > 0
lim
(x,y)→(0,0)
(x,y)=(0,0)
f(x, y) = lim
(x,y)→(0,0)
(x,y)=(0,0)
| sin(xy) − xy|
|xy| 3
α
|xy| 3α
(x2 + y2 1
=
) 3 6α |xy| ≤ 1
2 (x2 + y 2 )
0 ≤ |xy|α
x2 1
≤
+ y2 2α (x2 + y 2 ) α−1 .
α > 1
lim
(x,y)→(0,0)
(x,y)=(0,0)
|xy| α
x2 = 0, lim
+ y2 (x,y)→(0,0)
(x,y)=(0,0)
⎛
⎜
⎝ lim
(x,y)→(0,0)
(x,y)=(0,0)
f(x, y) = 0 = f(0, 0),
|xy| α
x2 + y2 α > 1 f α ≤ 1 y = mx
|xy| α
x2 |m|α
=
+ y2 m2 + 1
|x| 2α
,
x2 α < 1 x → 0 +∞ α = 1 |m|/(m 2 + 1) m
f f α > 1
lim
(x,y)→(0,0)
(x,y)=(0,0)
|xy| α
x2 =
+ y2 lim
ρ→0 +
x=ρ cos θ
y=ρ sin θ
|ρ2 sin θ cos θ| α
ρ 2 = lim
ρ→0 +
x=ρ cos θ
y=ρ sin θ
1
2 α ρ2α−2 | sin 2θ| α .
α > 1 θ α = 1 +∞ α < 1
f : R2 → R f(x, y) = (x2 + y2 )e−y2
C = {(x, y) ∈ R 2 : y ≥ 1 − x 2 |x| ≤ 1 y ≥ |x| − 1 |x| ≥ 1}.
lim f(x, y), lim
|(x,y)|→∞
|(x,y)|→∞
(x,y)∈C
f C
f
lim
|(x,y)|→∞
y=0
f(x, y) = lim
|x|→∞ x2 = +∞, lim
f(x, y).
f(x, y) = lim
|(x,y)|→∞
|y|→∞
x=0
y2e −y2
= 0.
lim f(x, y)
|(x,y)|→∞
0 ≤ f(x, y) ≤ (1 + y2 )e−y2 |x| ≤ 1 |x| > 1 (x, y) ∈ C y − 1 ≥ |x|
(y − 1) 2 ≥ x2 (x, y) ∈ C |x| > 1 0 ≤ f(x, y) ≤ ((y − 1) 4 + y2 )e−y2 (x, y) ∈ C
0 ≤ f(x, y) ≤ (1 + y 2 )e −y2
+ ((y − 1) 4 + y 2 )e −y2
.
|(x, y)| → ∞ (x, y) ∈ C y → +∞ 0
lim
|(x,y)|→∞
(x,y)∈C
f(x, y) = 0.
⎞
⎟
⎠
3