pdf (it, 2.898,97 KB, 25/01/13)

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 +∞
0 < α < 1 θ /∈ {0, π/2, π, 3π/2}
lim
(x,y)→(0,0)
1 − cos(xy)
log(1 + x 2 + y 2 )
lim
x→0
y→0
1 − cos(xy)
(xy) 2
(x2 + y2 )
log(1 + x2 + y2 ) (x2 + y2 )
(xy) 2
lim
(x,y)→(0,0)
1
=
2 lim
x→0
y→0
x3y2 x4 .
+ y6 (xy) 2
(x2 + y2 1
=
) 2 lim
ρ→0 +
ρ4 cos2 θ sin2 θ
ρ2 z = x 2 v = y 3 |x| = z 1/2 y = v 1/3
lim
(x,y)→(0,0)
x3y2 x4 + y6
= lim
(x,y)→(0,0)
z = ρ cos θ v = ρ sin θ
lim
x
3y2
(x,y)→(0,0)
x 4 + y 6
|x| 3y2 x4 + y6 = lim
ρ→0 + ρ3/2+2/3 | cos θ|3/2 sin 2/3 θ
ρ 2
= lim
z→0 +
v→0
|z| 3/2v2/3 z2 = lim
+ v2 (z,v)→(0,0)
|z| 3/2v2/3 z2 .
+ v2 = 0.
= lim
ρ→0 + ρ1/6 | cos θ| 3/2 sin 2/3 θ ≤ lim
ρ→0 + ρ1/6 = 0.
lim
(x,y)→(1,0)
y 2 log x
(x − 1) 2 + y 2 lim
(x,y)→(0,0)
sin(x 2 + y 2 )
x 2 + y 2
sin(x
lim
(x,y)→(0,0)
y=0
2 + y2 )
x2y2 + y4 x
lim
(x,y)→(0,0)
3 + x sin2 (y)
x2 + y2 xy(x
lim y arctan(y/x) lim
(x,y)→(0,0) (x,y)→(0,0)
2 − y2 )
x2 + y2 xy
lim
(x,y)→(0,0) x2 + xy + y2 xyz
lim
(x,y,z)→(0,0,0) x
(x,y)=(0,0)
2 + y2 (x
lim
|(x,y,z)|→+∞
2 + y2 ) 2
x2 + z2 lim
|(x,y,z)|→+∞
lim
|(x,y,z)|→+∞ x4 + y 2 + z 2 − x + 3y − z lim
|(x,y,z)|→+∞ x4 + y 2 + z 2 − x 3 + xyz − x + 4
(0, 0) 0
1
1
xz