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f : D → Rm D ⊆ Rn c D f
c
lim f(x) − f(c) = 0,
x→c
x∈D
j = 1, ..., m
f = (f1, ..., fm)
lim fj(x) = fj(c),
x→c
x∈D
R 2
(x, y) ∈ R x = ρ cos θ y = ρ sin θ
R 2 \ {0} ρ = x 2 + y 2 |(x, y)| = ρ f : D → R D ⊆ R 2
(0, 0) D
lim
(x,y)→(0,0)
(x,y)∈D
f(x, y) = lim
ρ→0 +
(ρ cos θ,ρ sin θ)∈D
f(ρ cos θ, ρ sin θ),
θ
θ
θ
(x, y) → (0, 0) f(x, y) = x 4 arctan y
f(ρ cos θ, ρ sin θ) = ρ 4 cos θ arctan(ρ sin θ)
| cos θ arctan(ρ sin θ)| ≤ π/2
0 ≤ |f(ρ cos θ, ρ sin θ)| ≤ π
2 ρ4 ,
ρ → 0 + 0 θ
0
R2
sin
f(x, y) =
arctan y
x , x = 0;
0, x = 0.
(x, y) x = 0 (0, 0)
lim sin arctan y
= lim sin (arctan tan θ) = sin θ
x ρ→0 +
(x,y)→(0,0)
x=0
θ f (0, 0) (0, ¯y) ¯y > 0
lim
(x,y)→(0,¯y)
x>0,y=¯y
lim
(x,y)→(0,¯y)
x=0
f(x, y) = 0
f(x, y) = lim f(x, ¯y) = 1,
x→0 +
(0, ¯y) ¯y > 0 (0, ¯y) ¯y < 0
lim
(x,y)→(0,¯y)
x>0,y=¯y
f(x, y) = lim f(x, ¯y) = −1,
x→0 +
(0, ¯y) ¯y < 0
f (x, y) x = 0