Universidad Tecnica Federico Santa MaraDepartamento de MatematicaCampus Santiago

Gua 2 - MAT-023

1. Determine, si existen, los siguientes lmites

(a) lim(x,y)(0,0)

| sen x sen y||x|+ |y|

(b) lim(x,y)(0,0)

x2y2

x2y2 + (y x)2

(c) lim(x,y)(0,0)

exy 1x2 + y2

(d) lim(x,y)(0,0)

xy sen(y3)

x4 + y4

(e) lim(x,y)(0,0)

sen(xy3)

x2 + y6

(f) lim(x,y)(0,0)

y3|x|

|x|+ y4

(g) lim(x,y)(0,0)

x3 + y2

|x|+ |y|

(h) lim(x,y)(0,0)

sen x+ sen y

x+ y

(i) lim(x,y)(1,0)

cos(xy) 1y

(j) lim(x,y)(1,0)

(x 1)4y2

(x 1)8 + y4

(k) lim(x,y)(0,0)

x3 + 8y3

x+ 2y

(l) lim(x,y)(0,0)

x2yx2 + y2

sen(xy)

(m) lim(x,y)(0,0)

|x|+ |y|sen(|x|+ |y|)

(n) lim(x,y)(0,0)

x2y

x2 + y

2. Determine para cuales valores de N existe el siguiente lmite:

lim(x,y)(0,0)

xy

x2 + y2

3. Determine para cuales valores de N {0} existe el siguiente lmite:

lim(x,y)(0,0)

x2y

x2 + y2

4. Sea f : A R donde,

A = {(x, y) R2 : |y| < x2} y f(x, y) = sen(|y|)|x|+ |y|

Determine, si existe, lim(x,y)(0,0)

f(x, y)

5. Sea f : A R donde,

A = {(x, y) R2 : |y| > x} y f(x, y) = xy|x|+ |y|

(a) Analice: limx0

(limy0

f(x, y)

)y lim

x0+

(limy0

f(x, y)

)(b) Demostrar que: lim

(x,y)(0,0)f(x, y) = 0

6. Analizarlim

(x,y,z)(0,0,0)

xyz

x2y2 + xyz + z2

7. Sea f(x, y) =

|x|

|x|+ |y|si |y| < x2

1 si (x, y) = (0, 0)

Existe lim(x,y)(0,0)

f(x, y)?

8. Sea f(x, y) =

x+ y si |x|+ |y| 2

1 si |x|+ |y| < 2Existe lim

(x,y)(1,1)f(x, y)?