x2+y2 x2−3x+ 2 (g) lim (x,y)→(0,0) x3y3 2x2+y2 1 (2)(h) f(x, y

MAT208 - Exerc´ıcios LISTA 4 - 2011

1. Determine o limite se existir, ou mostre que o limite n˜ao existe (justifique sua resposta):

(a) lim

(x,y)→(5,−2)(x⁵ + 4x³y−5xy²) (b) lim

(x,y)→(0,0)

3x²−y²+ 5 x²+y²+ 2 (c) lim

(x,y)→(0,0)

x³+xy²

x²+y² (d) lim

(x,y)→(1,1)

x−y

x³−y³ (e) lim

(x,y)→(0,0)

x³y³ 2x²+y² (f) lim

(x,y)→(0,0)

8x²y²

x⁴+y⁴ (g) lim

(x,y)→(0,0)

8x⁴y x⁴+y⁴ (h) lim

(x,y)→(0,0)

xy³

x²+y⁶ (i) lim

(x,y)→(0,0)

xy px²+y² (j) lim

(x,y,z)→(3,0,1)e^−xy sen (πz

2 ) (k) lim

(x,y,z)→(0,0,0)

x²+ 2y²+ 3z² x²+y²+z² (l) lim

(x,y,z)→(0,0,0)

xyz²

x²+y² +z² (m) lim

(x1,x2,x3,x4)→(1,1,1,1)(x₁² +x₂²+x₃²+x₄²)

2. Calcule o limite:

(a) lim

(x,y)→(6,3)xycos(x−2y) (b) lim

(x,y)→(π,2π)

cosy+ 1

y−sinx (c) lim

(x,y)→(1,1)ln|x²y²−2|

(d) lim

(x,y)→(1,1)lnx³−y³

x−y (e) lim

(x,y)→(2,2)

x+y−4

√x+y−2 (f) lim

(x,y)→(0,0)

x²+y² px²+y²+ 1−1 (g) lim

(x,y)→(0,0)

sin 2x

x(y²−1) (h) lim

(x,y)→(0,0)

(xy−2) tany² y²

3. Determine o maior conjunto no qual a fun¸c˜ao ´e cont´ınua:

(a) f(x, y) = 1

x²−y (b) f(x, y) = ln(2x+ 3y) (c) g(x, y) = arctan(x+√ y)

(d) h(x, y) = sin 1

xy (e) f(x, y) = x+y

2 + cosx (f) g(x, y) = x²+y² x²−3x+ 2 (g) lim

(x,y)→(0,0)

x³y³ 2x²+y²

1

(h) f(x, y) =

( x³y³

2x²+y² se(x, y)6= (0,0) 1 se(x, y) = (0,0)

(i)f(x, y) =

( x³y³

2x²+y² se(x, y)6= (0,0) 0 se(x, y) = (0,0) (j) g(x, y, z) = xyz

x²+y²−z (k)f(x, y, z) = xyz px²+y²−z (l)f(x, y, z) = ln(x²+y²+z²) + z²

x²+y² (m) f(x, y, z, t) = x² +y² x+y+z−t 4. Calcule o vetor gradiente de f em cada ponto (x, y) se

(a) f(x, y) =x⁵+ 3x³y²+ 3xy⁴ (b) f(x, y) = (xy+ 1)² (c) f(x, y) =e^−xsen(x+y)

5. Mostre que as derivadas parciais de f(x, y) =

xy

x²+y² se(x, y)6= (0,0) 0 se(x, y) = (0,0)

existem em (0,0) e calcule-as. Mas mostre tambem que f n˜ao ´e cont´ınua em (0,0).

Algumas respostas 1. (a) 2025 (b) ⁵₂ (c) 0 (d) ¹₃ e) 0 (f) n˜ao existe

(g) 0 (h) n˜ao existe i) 0 (j) 1 (k) n˜ao existe (l) 0 (m) 4 2. (a) 18 (b) _π¹ (c) 0 (d) ln 3 (e) 4 (f) 2 (g) −2 (h) −2 3. (a) {(x, y)|y 6=x²} (b) {(x, y)|y >−²₃x} (c) {(x, y)|y >0}

(d) {(x, y)|x6= 0, y 6= 0} (e) R² (f) {(x, y)|x6= 1, x6= 2} (g) R² − {(0,0)} (h) R²− {(0,0)}

(i)R² (j) {(x, y, z)∈R³ | z 6=x²+y²}

(k) {(x, y, z)| z < x²+y²}(abaixo do paraboloide z =x²+y²)

(l){(x, y, z) | x²+y² 6= 0}=R³−eixo 0z (m) {(x, y, z, t)∈R⁴ | x+y+z 6=t}

4. (b) ∇(x, y) = (2y(xy~ + 1),2x(xy+ 1)) 5. ^∂f_∂x(0,0) = 0, ^∂f_∂y(0,0) = 0

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