16.2. EXERCISES 287

16.2 Exercises1. Sketch the contour graph of the function of two variables f (x,y)= (x−1)2+(y−2)2.

2. Which of the following functions could correspond to the following contour graphs?z = x2 +3y2,z = 3x2 + y2,z = x2− y2,z = x+ y.

-4 -2 0 2 4-4

-2

0

2

4

-4 -2 0 2 4-4

-2

0

2

4

-4 -2 0 2 4-4

-2

0

2

4

3. Which of the following functions could correspond to the following contour graphs?z = x2−3y2,z = y2 +3x2,z = x− y,z = x+ y.

-4 -2 0 2 4-4

-2

0

2

4

-4 -2 0 2 4-4

-2

0

2

4

-4 -2 0 2 4-4

-2

0

2

4

4. Which of the following functions could correspond to the following contour graphs?z = sin(x+ y),z = x+ y,z = (x+ y)2,z = x2− y.

-4 -2 0 2 4-4

-2

0

2

4

-4 -2 0 2 4-4

-2

0

2

4

-4 -2 0 2 4-4

-2

0

2

4

5. Find the following limits if they exist. If they do not exist, explain why.

(a) lim(x,y)→(0,0)x2−y2

x2+y2

(b) lim(x,y)→(0,0)2x3+xy2−x2−2y2

x2+2y2

(c) lim(x,y)→(0,0)sin(x2+y2)

x2+y2

(d) lim(x,y)→(0,0)sin(x2+2y2)

x2+2y2

(e) lim(x,y)→(0,0)sin(x2+2y2)

2x2+y2

(f) lim(x,y)→(0,0)(x2−y4)

2

(x2+y4)2

6. Find the following limits if they exist. If they do not exist, tell why.

(a) lim(x,y)→(0,0) x (x2−y4)

2

(x2+y4)2

(b) lim(x,y)→(0,0)xsin(x2+2y2)

2x2+y2

(c) lim(x,y)→(0,0)xy

x2+y2