limit. 1. lim (x,y) (1,3) (4xy2-x) xy³ 2. lim 4x - y (x,y) (0.0) sin y-1 1-6 Use limit laws and continuity properties to evaluate the 17. 18. lim xz² (x,y,z) (2.-1.2) √x2 + y² + z² lim In(2x+y-z) (x,y,z) (2,0,-1) 3. lim 4. (x,y) (-1,2)x+y lim (x,y) (1,-3) e2x-y2 sin(x² + y²+z²) 19. lim (x,y,z) (0,0,0) x² + y²+z² 5. lim In(1+x²y³) 6. lim x√y +2x sin (x,y) (0,0) + y²+z² (x,y) (4,-2) 20. lim (x,y,z) (0,0,0) x² + y² + z² 7-8 Show that the limit does not exist by considering the limits as (x, y)(0, 0) along the coordinate axes. √x²+y²+z² 21. lim (x,y,z) (0,0,0) + y²+ 3 7. (a) lim (b) (x,y) (0,0) x2+2y2 8. (a) lim x-y (x,y) (0,0) x² + y² (b) lim x+y lim (x,y) (0,0) 2x2 + y² cosxy (x,y)(0,0) x² + y² 1 -1 22. lim tan (x,y,z) (0,0,0) x² + y² + z² 9-12 Evaluate the limit using the substitution z = x² + y² and observing that z→0+ if and only if (x, y)→ (0,0). ■ 23. lim 23-26 Evaluate the limits by converting to polar coordinates, as in Example 7. ■ √√x² + y² ln(x² + y²) 9. lim (x,y) (0,0) 11. lim (x,y) (0,0) sin(x² + y²) x² + y² e-1/(x²+y²) 10. lim (x,y)→ (0,0) 1 - cos(x² + y²) x² + y² (x,y) (0,0) x2 y2 24. lim (x,y) (0,0) y In(x² + y²) 25. lim (x,y)→ (0,0) 12. lim (x,y)→ (0,0) x² + y² xy 26. lim 13-22 Determine whether the limit exists. If so, find its value. (x,y) (0.0) x2+2y2 27-28 Evaluate the limits by converting to spherical coordi- nates (p, 0, 0) and by observing that p→0+ if and only if (x, y, z) (0, 0, 0). ■ 13. lim x4 - y4 (x,y) (0.0) x2 + y² x4 - 16y4 14. lim (x,y) (0.0) x2+4y2 15. lim xy (x,y) (0,0) 3x²+2y2 16. lim 1 x2 y2 (x,y) (0.0) x2 + y2 27. xyz lim (x,y,z) (0,0,0) x² + y²+z²