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(0,0)
f(x, y) = x3
y
f(x, y) = x+2y
x2−y2
f(x, y) = x2+y2
|x|+|y|
(0,0)
f(x, y) = x3+y3
x2+y2
f(x, y) = xy
x4+y4
f(x, y) = x2y
x4+y2
f(x, y) = xy
x−y
(0,0)
f(x, y) = sin xy
√x2+y2
f(x, y) = 1−cos(xy)
xy2
f(x, y) = xy= eyln x
f(x, y) = sh xsh y
x+y
f:R+×R∗
+→Rf(x, y) = xyx > 0f(0, y)=0
f
R+×R+
f:R→RC1F:R2\ {(0,0)} → R
F(x, y) = f(x2+y2)−f(0)
x2+y2
lim(x,y)→(0,0) F(x, y)

f(0,1/n)→0f(1/n, 1/n3)→1 (0,0)
f(0,−1/n)=2n→+∞f(0,1/n) = −2n→ −∞ (0,0)
0≤f(x, y)≤x2+2|x||y|+y2
|x|+|y|=|x|+|y| → 0f(rcos θ, r sin θ) = O(r)
x=rcos θ y =rsin θ r =px2+y2→0
f(x, y) = r(cos3θ+ sin3θ)−→
(x,y)→(0,0) 0
f(1/n, 0) →0f(1/n, 1/n3)→1f(0,0)
f(1/n, 0) = 0 →0f(1/n, 1/n2)=1/2→1/2f
(0,0)
f(1/n, 0) = 0 →0f(1/n + 1/n2,1/n) = 1/n2+1/n3
1/n2→1f
(0,0)
|f(x, y)| ≤ |xy|
√x2+y2=r|sin θcos θ| −→
(x,y)→(0,0) 0
f(x, y) = x1−cos(xy)
x2y2limt→01−cos t t2=1 2f(x, y)−→
(x,y)→(0,0) 0
f(1/n, 0) →1f(1/n, 1/ln n)→1/e (0,0)
x→0f(x, −x+x3)∼ −1
xf(0,0)
f(x, y) = exp(yln x)R∗
+×R∗ +
(0, b)b > 0
(x, y)→(0, b) (x, y)∈R∗
+×R∗ +yln x→ −∞
f(x, y) = xy→0
(0, y)→(0, b)f(x, y) = 0 →0
f(0, b)
fR+×R+
f(0,0) = limy→0f(0, y) = 0
f(0,0) = limx→0f(x, x)=1
cx,y ∈]0 ; x2+y2[
F(x, y) = f0(c)
(x, y)→(0,0) cx,y →0F(x, y)→f0(0)