SECTION 14.2 LIMITS AND CONTINUITY
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24.
25– 26 Find and the set on which is
continuous.
,
26. ,
;27– 28 Graph the function and observe where it is discontinuous.
Then use the formula to explain what you have observed.
27.
29– 38 Determine the set of points at which the function is
continuous.
29. 30.
31. 32.
33. 34.
35.
36.
38.
39– 41 Use polar coordinates to find the limit. [If are
polar coordinates of the point with , note that
as .]
40.
41. lim
共x, y兲 l 共0, 0兲 e⫺x2⫺y2⫺1
x2⫹y2
lim
共x, y兲 l 共0, 0兲 共x2⫹y2兲 ln共x2⫹y2兲
lim
共x, y兲 l 共0, 0兲 x3⫹y3
x2⫹y2
39.
共x, y兲l共0, 0兲
rl0⫹
r艌0共x, y兲
共r,
兲
f共x, y兲苷
再
0
xy
x2⫹xy ⫹y2if
if
共x, y兲苷共0, 0兲
共x, y兲苷共0, 0兲
f共x, y兲苷
再
1
x2y3
2x2⫹y2if
if
共x, y兲苷共0, 0兲
共x, y兲苷共0, 0兲
37.
f共x, y, z兲苷sx⫹y⫹z
f共x, y, z兲苷sy
x2⫺y2⫹z2
G共x, y兲苷tan⫺1
(
共x⫹y兲⫺2
)
G共x, y兲苷ln共x2⫹y2⫺4兲
F共x, y兲苷ex2y⫹sx⫹y2
F共x, y兲苷arctan
(
x⫹sy
)
F共x, y兲苷x⫺y
1⫹x2⫹y2
F共x, y兲苷sin共xy兲
ex⫺y2
f共x, y兲苷1
1⫺x2⫺y2
28.
f共x, y兲苷e1兾共x⫺y兲
f共x, y兲苷1⫺xy
1⫹x2y2
t共t兲苷t⫹ln t
f共x, y兲苷2x⫹3y⫺6t共t兲苷t2⫹st
25.
hh共x, y兲苷t共f共x, y兲兲
lim
共x, y兲 l 共0, 0兲 xy3
x2⫹y6
1. Suppose that . What can you say
about the value of ? What if is continuous?
2. Explain why each function is continuous or discontinuous.
(a) The outdoor temperature as a function of longitude,
latitude, and time
(b) Elevation (height above sea level) as a function of longi-
tude, latitude, and time
(c) The cost of a taxi ride as a function of distance traveled
and time
3–4 Use a table of numerical values of for near the
origin to make a conjecture about the value of the limit of
as . Then explain why your guess is correct.
3. 4.
5–22 Find the limit, if it exists, or show that the limit does
not exist.
5. 6.
7. 8.
10.
11. 12.
14.
15. 16.
17. 18.
19.
20.
22.
;23– 24 Use a computer graph of the function to explain why the
limit does not exist.
23. lim
共x, y兲 l 共0, 0兲 2x2⫹3xy ⫹4y2
3x2⫹5y2
lim
共x, y, z兲 l 共0, 0, 0兲 yz
x2⫹4y2⫹9z2
lim
共x, y, z兲 l 共0, 0, 0兲 xy ⫹yz2⫹xz2
x2⫹y2⫹z4
21.
lim
共x, y, z兲 l 共0, 0, 0兲 x2⫹2y2⫹3z2
x2⫹y2⫹z2
lim
共x, y, z兲l共3, 0, 1兲 e⫺xy sin共
z兾2兲
lim
共x, y兲 l 共0, 0兲 xy4
x2⫹y8
lim
共x, y兲 l 共0, 0兲 x2⫹y2
sx2⫹y2⫹1⫺1
lim
共x, y兲 l 共0, 0兲 x2 sin2y
x2⫹2y2
lim
共x, y兲 l 共0, 0兲 x2yey
x4⫹4y2
lim
共x, y兲 l 共0, 0兲 x4⫺y4
x2⫹y2
lim
共x, y兲 l 共0, 0兲 xy
sx2⫹y2
13.
lim
共x, y兲 l 共0, 0兲 6x3y
2x4⫹y4
lim
共x, y兲 l 共0, 0兲 xy cos y
3x2⫹y2
lim
共x, y兲 l 共0, 0兲 x2⫹sin2y
2x2⫹y2
lim
共x, y兲 l 共0, 0兲 y4
x4⫹3y4
9.
lim
共x, y兲 l 共1, 0兲 ln冉1⫹y2
x2⫹xy冊
lim
共x, y兲 l 共2, 1兲 4⫺xy
x2⫹3y2
lim
共x, y兲 l 共1, ⫺1兲 e⫺xy cos共x⫹y兲lim
共x, y兲 l 共1, 2兲 共5x3⫺x2y2兲
f共x, y兲苷2xy
x2⫹2y2
f共x, y兲苷x2y3⫹x3y2⫺5
2⫺xy
共x, y兲l共0, 0兲
f共x, y兲
共x, y兲f共x, y兲
ff 共3, 1兲
lim共x, y兲l共3, 1兲 f共x, y兲苷6
EXERCISES
14.2
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