|
|
?????????????
??1
1. ?1????2????3???4???
2. ?1?????????????????????????????????????
?2??????????????????????????????????????????????????3???????
?3???????????????????????????????
3. ??{1}?{2}?{3}?{1,2}?{1,3}?{2,3}?{1,2,3}?
4. ?1?{0,1,2,3,4} ?2?{3,4} ?3?{(?1,?1),(0,0),(1,1)}
5. ?1?{x|x?2?3,x?Z} ?2?{x|x?x?12?0} ?3?{(x,y)|y?x,y?x}
6. ?1?{1,3} ?2?{1,2,3,5} ?3?? ?4?{1,2,3,4,5,6} ?5?{2} ?6??
?7?{4,5,6} ?8?{1,3,4,5,6} ?9?{1,2,3,4,5,6} ?10?{4,6}
7. 23
A?A?B?B?A?(A?B)?B
?((A?A)?(A?B))?B
?(??(A?B))?B
?(A?B)?B
?(A?B)?(B?B)
?(A?B)?U
?A?B
8. ?1?(?5,5) ?2?(?2,0) ?3?(??,?3]?[1,??) ?4?(1,2]
?5?[4,??) ?6?(??,4)
9. ?1?A?B?{1}?A?B?[0,3]?A?B?[0,1)?
?2?A?B?[2,4]?A?B?[?1,4]?A?B?[?1,2)?
10. ?1?(,)?2?(,2)?(2,)?
11. ?1????????? ?2????????? ?3?????????
?4??????????[?1,1]??y??x??x?y??x2
12. ?1?(??,?2)?(?2,2)?(2,??) ?2?(??,?1]?[1,??) ?3?(?1,1] 35223252
?4?(??,??)?5?(?2,2)?6?[1,5]
?7?(?
2?2k?,?
2?2(k?1)?)?k?0,?1,?2,? ?8?(?2,?1)?(?1,1)?(1,??)
?9?(??,?2)?(3,??) ?10?[2,4]
13?1?f(0)?02?3?0?5??5?f(1)?12?3?1?5??1?
f(?1)?(?1)2?3?(?1)?5??7?f(?x)?(?x)2?3?(?x)?5?x2?3x?5? f()?()?3?1
x1x2113?5?2??5? xxx
14. f(x)?f(x?1?1)?(x?1)2?2(x?1)?3?x2?4?
f(x?1)?(x?1)2?4?x2?2x?3? sin(?)??2?f(0)?0?1?1?f(?)???1??? 15. f(?)?2222??2?
x2x2x2?116. ?x?D?(??,??)??f(x)?1???1??1??2? 2221?x1?x1?x
17. ?1????? ?2?(??,2]??????[2,??)????? ?3?(??,1]?????[1,??)????? ?4????? ?5?(??
2?k?,?
2?k?)?k?0,?1,?2,???
????? ?6?????
18. ?1???? ?2??????? ?3???? ?4???? ?5???????
?6???? ?7??????? ?8???? ?9???? ?10????
19. ?1?????????x???F(?x)?
???
?2?????????x??G(?x)?
??G(x)?????
20. ?1?? ?2?2? ?3?? ?4?2?
21. ?1????x?(??,??)??f(x?2)?f(x)?f(2)????x??1???1[f(?x)?f(x)]?F(x)???F(x)??211[f(?x)?f(x)]??[f(x)?f(?x)]??G(x)?22f(1)?f(?1)?f(2)????f(x)?(??,??)????????f(?1)??f(1)???f(2)?2f(1)?2a??f(5)?f(3)?f(2)?(f(1)?f(2))?f(2)?f(1)?2f(2)???
f(5)?5a?
?2???f(x)??2???????????f(x?2)?f(x)????
f(x?2)?f(x)?f(2)???f(2)?0???1??f(2)?2a???a?0?
222. ?1?y?arcsinu?u?1?x ?2?y??u?lnv?v?x
w?2?3?y?u?u?2?v?v?cosx ?4?y?eu?u?arctanv?v?
w?1?x
23. ?1?y?1?x1bx? ?2?y?ex?1 ?3?y?x?2 ?4?y??x??1? 1?xkk
24. ?1?? ?2?? ?3?? ?4???
??2
1. (1) 0 (2) 1 (3) 0 (4) 0
2?(1)3 (2)2 (3)0 (4) ?? (5) ?
3.???????????????????
1limn??nn21??? n2limnn??
4. ??????? 1(1)y?xcos?x?0?????? x
??????0,????,?x??,xcos
(2)y?1?x?x??1?????. x1?x?? x
????M?0,???M?1,?x??,
5. ??????
?1?1?2?0
6. ???????
?1?0?2?1 2x?111????1?M?1?1?M xxx
?3?2?4?1 2
7. ???????
?1?4 ?2??
1?3?2?4? 3
1?5???6? 4
?7?-1?8?
?x?1,x?0?x?0???????x?0??????? 8. ?f(x)??0,
?x?1,x?0?
??lim-f(x)??1,lim-f(x)?1,f(0)?0???f(x)?x?0?????? x?0x?0
x2?ax?b?5??a?b 9. ??limx?11?x
x2?ax?b?5?????????a?b?1?0,??a??b?1,??limx?11?x
x2?(b?1)x?b(x?b)(x?1)lim?lim?1?b?5??b?6,a??7 x?1x?11?x1?x
10. ???????
lim?x?cosx?lim?x?0x12x2?1?x?0?2
tanx?sinx1?cosxx21?lim?lim??2?lim x?0x?0x2?cosxx?02x2?cosx2x3
tan2x22x2
?lim2?2 ?3?lim2x?0x?0xx
?4?limx?02??cosx1?cosxsinx2?lim?lim?x?08 sin2x(2??cosx)sin2xx?022?2sinxcosx
xx?1?11??x?2???5?lim??lim1????x??x?1x?????x?1??e
1?x??6?lim??? x??x?1e??
1??3?x???7?lim???lim?1??x??2?xx??x?2????
??8?lim?1?x?0?x??2?x?1xxx?(x?2)?21? e?e ?1
2
?x2???e?9?lim?x???x2?1???x
?10?limsinx?sin1cosx?lim?cos1 x?1x?1x?11
1?cosxx2
?lim?0?11?limx?0x?02sinxsinx
lim(1?x)tan?x
2?12?x?1?lim(1?x)x?1?12?lim?xx?1x? cos?sin222sin?x
sin3x?2sinx3? ?13?lim2?54?sinxx?2
?14? lim
x?cotxx??
22??1
111????????? 1?21?221?2n
111111???????????? 2n2n1?21?2221?2211. ?????xn?
11??112. ???limn?2?2???2?? n??n??n?2?n?n???
???n?n11?n?n?1 ?n??????2?2222n??n?n??n?n??n??n?2?
2n
13. ?lim? n??n!
2n2? ???n????n!n
????????????
???
1. ??????????????????????????????????????????? (1) {(x,y)|x?0};
(2) {(x,y)|1?x2+y2<4};
(3) {(x,y)|y<x2};
(4) {(x,y)|(x-1)2+y2?1}?{(x,y)|(x+1)2+y2?1}.
??(1)???????????R2????{(x,y)|x=0}. (2)????????????, ????{(x,y)|1?x2+y2?4}? ???{(x,y)|x2
+y2
=1}?{(x,y)| x2
+y2
=4}. (3)?????????? ????{(x,y)|y?x2
}? ???{(x,y)| y=x2}.
(4)????????????????
???{(x,y)|(x-1)2+y2=1}?{(x,y)|(x+1)2+y2=1}. 2. ??f(x,y)=x2+y2-xytan
xy
,??f(tx,ty).
??f(tx,ty)?(tx)2?(ty)2?tx?tytan
tx2
ty
?tf(x,y).
3. ??f(u,v,w)?uw?wu?v,??f(x?y,x?y,xy). ??f(x+y, x-y, xy) =(x+y)xy+(xy)x+y+x-y =(x+y)xy+(xy)2x. 4. ???????????
(1)z?ln(y2
?2x?
1);
(2)z?
?
(3)z?
ln(1?x?
y)
(4)u?
(5)z?
(6)z?ln(y?x)?
(7)u?arccos
??(1)D?{(x,y)|y2
?2x?1?0}.
(2)D?{(x,y)|x?y?0,x?y?0}.
(3)D?{(x,y)|4x?y?0,1?x?y?0,x?y?0}.
2
2
2
2
2
(4)D?{(x,y,z)|x?0,y?0,z?0}. (5)D?{(x,y)|x?0,y?0,x2
?y}. (6)D?{(x,y)|y?x?0,x?0,x2
?y2
?1}. (7)D?{(x,y,z)|x2
?y2
?0,x2
?y2
?z2
?0}.5. ???????
y
(1)lim
ln(x?e)x?1 y?0
(3)lim
x?0xy
y?0
(5)lim
sinxyx?0x
; y?0
??(1)??
?ln2.(2)??=+?. (3)??
=lim
1x?0??
y?0
4
.
(4)??
=lim
x?0xy?1?1
?2.
y?0
(5)??=lim
sinxyx?0xy
?y?1?0?0.
y?0
1
(x2?y2)
2
2(6)??=lim
y2
x?02
2
2
x?0y?0
(x?y)e
x?y
2
?lim
x?2e
(x2
?y2
)
?0.
y?0
6. ?????????O(0?0)?????? ?sin(x3?2
2
(1)z??
y3)
?y2
,x?y?0,?x2
??
0,x2
?y2
?0;
(2)lim
1x?0x2
?
y
2
;
y?0
(4)lim
x?0
y?0
2
2
(6)lim
1?cos(x?y)x?0(x2
?y2
.y?0
)e
x2
?y
2
?sin(x3?y3)
,?
(2)z??x3?y3
?0,?
x?y?0,x?y?0;
3
3
33
22
?xy
,?222
(3) (2)z??xy?(x?y)
?0,?
x?y?0,x?y?0;
2
2
22
??(1)??0?
sin(x?y)x?y
2
2
33
?
x?yx?y
3
3
332
2
?
sin(x?y)x?y
3
3
33
?(x?y)
sin(x?y)x?y
3
3
33
?lim(x?y)?0??lim
x?0
y?0
sin(x?y)x?y
3
3
x?0y?0
?lim
sinuu
u?0
?1,
?limz?0?z(0,0).
x?0y?0
????O(0,0)???. (2)limz?lim
x?0y?0
sinuu
u?0
?1?z(0,0)?0
?O(0,0)?z????.
(3)?P(x,y) ???y=x??(0?0)???
limz?lim
x?x
2
22
2
x?0y?x?0
x?0
x?x?0
?1,
??P(x,y) ???y=-x??(0?0)???
limz?lim
x(?x)
2
22
2
2
x?0
y??x?0
x?0
x?(?x)?4x
?lim
x
2
2
x?0
x?4
?0
?limz???.???z?O(0?0)????.
x?0y?0
7. ???????????? (1) f(x,y)=
x?y
3
23
x?y
; (2) f(x,y)=
y?2xy?2x
2
2
2
;
(3) f(x,y)=ln(1?x2?y2);
?x?x2?2ey,
(4)f(x,y)=?y
?
0,?
y?0,y?0.
??(1)???y=-x???????????????y=-x?????????????
?????.
(2)???y2=2x????????????????y2=2x????????.??????????.
(3)???x2+y2=1???????????????x2+y2=1???????.??????
????.
(4)???P(x,y)???y=x??O(0,0)?.
limf(x,y)?lim
xx
2
x?0x?0
e
?1
??.
y?x?0
?(0?0)??????????????????. 8. ?????????? 22
(1)z=x2
y+
xy
2
; (2)s=
u?vuv;
(3)z=x
ln;
(4)z=lntanxy
;
(5)z=(1+xy)y; (6)u=zxy;
y
(7)u=arctan(x-y)z
; (8)u?xz.
??(1)
?z?2xy?
1?z2
2x?x
y
2
,
?y
?x?
y
3
.
(2)s?u?
v
?s1uv
u
?u
?v
?
vu
2
,
?s?v
??v
2
?
1u
.
2
(3)?z?x
?lnx2x?
122
2
ln(x?y)?
x
x2
?y
2
,?z?y
?xy?
xy2x2
?y
2
.
(4)?z?
1
?x
?sec
2
xtan
xy?1y?2ycsc2xy
, y
?zx2xx?y
?
1tan
x?sec
2
y
?(?
xy
2
)??y
2
csc
2y
.
y
(5)??????lnz?yln(1?xy)
?
?z?x
?(1?xy)y
??yln(1?xy)??y
2
x?(1?xy)y
?
1?xy
?y2(1?xy)
y?1
.
?z
y
?y?(1?xy)??yln(1?xy)??y?y?(1?xy)?
ln(1?xy)?yx??1?xy??
?(1?xy)y?
?xy??ln(1?xy)?
1?xy?.?(6)
?u?x?lnz?z
xy
?y
?uxy?1
?y
?lnz?z
xy
?x
?u?z
?xy?z
(7)?uz(x?y)
z?1?x?
11?[(x?y)z]2
?z(x?y)
z?1
?
1?(x?y)
2z
.
?uz(x?y)
z?1
(?1)
?y)
z?1?y?
?1?[(x?y)z]
2
??
z(x1?(x?y)
2z
.
?uz?y)z
?z
?
(x?y)ln(x?y)ln(x?y)1?[(x?y)z
]
2
?
(x1?(x?y)
2z
.
y
(8)
?uz
?1
?x?
yz
x
. ?uy
?y?xzlnx?
1?1y
z
z
xzlnx.
?u
y
y
?xzlnx????y?
yz?z?z2??
??z2xlnx.29.??u?
xy
2
?ux?y
????x
?x
?y
?u?y
?3u.
2
???
?u(x?y)?x2y
2
222xy3
?x?
2xy(x?y)2
?
xy?(x?y)
2
.
?ux2
y2
?2yx3
??????y
?(x?y)2
.
x
?u?ux2
y2
?? (x?y)?x
?y?y
?
3(x?y)
2
?3u.
??1?110.?z?e??
?xy??
????x
2
?zz?x
?y
2
??y
?2z.
??1
1?????1?1??? ?z?e
?xy??
??1??
1
?????xy??
?x
????x2????x2e,
?
?z??x,y?????
?????????????-??1-??4
??1
1. ?1????2????3???4???
2. ?1?????????????????????????????????????
?2??????????????????????????????????????????????????3???????
?3???????????????????????????????
3. ??{1}?{2}?{3}?{1,2}?{1,3}?{2,3}?{1,2,3}?
4. ?1?{0,1,2,3,4} ?2?{3,4} ?3?{(?1,?1),(0,0),(1,1)}
5. ?1?{x|x?2?3,x?Z} ?2?{x|x?x?12?0} ?3?{(x,y)|y?x,y?x}
6. ?1?{1,3} ?2?{1,2,3,5} ?3?? ?4?{1,2,3,4,5,6} ?5?{2} ?6??
?7?{4,5,6} ?8?{1,3,4,5,6} ?9?{1,2,3,4,5,6} ?10?{4,6}
7. 23
A?A?B?B?A?(A?B)?B
?((A?A)?(A?B))?B
?(??(A?B))?B
?(A?B)?B
?(A?B)?(B?B)
?(A?B)?U
?A?B
8. ?1?(?5,5) ?2?(?2,0) ?3?(??,?3]?[1,??) ?4?(1,2]
?5?[4,??) ?6?(??,4)
9. ?1?A?B?{1}?A?B?[0,3]?A?B?[0,1)?
?2?A?B?[2,4]?A?B?[?1,4]?A?B?[?1,2)?
10. ?1?(,)?2?(,2)?(2,)?
11. ?1????????? ?2????????? ?3?????????
?4??????????[?1,1]??y??x??x?y??x2
12. ?1?(??,?2)?(?2,2)?(2,??) ?2?(??,?1]?[1,??) ?3?(?1,1] 35223252
?4?(??,??)?5?(?2,2)?6?[1,5]
?7?(?
2?2k?,?
2?2(k?1)?)?k?0,?1,?2,? ?8?(?2,?1)?(?1,1)?(1,??)
?9?(??,?2)?(3,??) ?10?[2,4]
13?1?f(0)?02?3?0?5??5?f(1)?12?3?1?5??1?
f(?1)?(?1)2?3?(?1)?5??7?f(?x)?(?x)2?3?(?x)?5?x2?3x?5? f()?()?3?1
x1x2113?5?2??5? xxx
14. f(x)?f(x?1?1)?(x?1)2?2(x?1)?3?x2?4?
f(x?1)?(x?1)2?4?x2?2x?3? sin(?)??2?f(0)?0?1?1?f(?)???1??? 15. f(?)?2222??2?
x2x2x2?116. ?x?D?(??,??)??f(x)?1???1??1??2? 2221?x1?x1?x
17. ?1????? ?2?(??,2]??????[2,??)????? ?3?(??,1]?????[1,??)????? ?4????? ?5?(??
2?k?,?
2?k?)?k?0,?1,?2,???
????? ?6?????
18. ?1???? ?2??????? ?3???? ?4???? ?5???????
?6???? ?7??????? ?8???? ?9???? ?10????
19. ?1?????????x???F(?x)?
???
?2?????????x??G(?x)?
??G(x)?????
20. ?1?? ?2?2? ?3?? ?4?2?
21. ?1????x?(??,??)??f(x?2)?f(x)?f(2)????x??1???1[f(?x)?f(x)]?F(x)???F(x)??211[f(?x)?f(x)]??[f(x)?f(?x)]??G(x)?22f(1)?f(?1)?f(2)????f(x)?(??,??)????????f(?1)??f(1)???f(2)?2f(1)?2a??f(5)?f(3)?f(2)?(f(1)?f(2))?f(2)?f(1)?2f(2)???
f(5)?5a?
?2???f(x)??2???????????f(x?2)?f(x)????
f(x?2)?f(x)?f(2)???f(2)?0???1??f(2)?2a???a?0?
222. ?1?y?arcsinu?u?1?x ?2?y??u?lnv?v?x
w?2?3?y?u?u?2?v?v?cosx ?4?y?eu?u?arctanv?v?
w?1?x
23. ?1?y?1?x1bx? ?2?y?ex?1 ?3?y?x?2 ?4?y??x??1? 1?xkk
24. ?1?? ?2?? ?3?? ?4???
??2
1. (1) 0 (2) 1 (3) 0 (4) 0
2?(1)3 (2)2 (3)0 (4) ?? (5) ?
3.???????????????????
1limn??nn21??? n2limnn??
4. ??????? 1(1)y?xcos?x?0?????? x
??????0,????,?x??,xcos
(2)y?1?x?x??1?????. x1?x?? x
????M?0,???M?1,?x??,
5. ??????
?1?1?2?0
6. ???????
?1?0?2?1 2x?111????1?M?1?1?M xxx
?3?2?4?1 2
7. ???????
?1?4 ?2??
1?3?2?4? 3
1?5???6? 4
?7?-1?8?
?x?1,x?0?x?0???????x?0??????? 8. ?f(x)??0,
?x?1,x?0?
??lim-f(x)??1,lim-f(x)?1,f(0)?0???f(x)?x?0?????? x?0x?0
x2?ax?b?5??a?b 9. ??limx?11?x
x2?ax?b?5?????????a?b?1?0,??a??b?1,??limx?11?x
x2?(b?1)x?b(x?b)(x?1)lim?lim?1?b?5??b?6,a??7 x?1x?11?x1?x
10. ???????
lim?x?cosx?lim?x?0x12x2?1?x?0?2
tanx?sinx1?cosxx21?lim?lim??2?lim x?0x?0x2?cosxx?02x2?cosx2x3
tan2x22x2
?lim2?2 ?3?lim2x?0x?0xx
?4?limx?02??cosx1?cosxsinx2?lim?lim?x?08 sin2x(2??cosx)sin2xx?022?2sinxcosx
xx?1?11??x?2???5?lim??lim1????x??x?1x?????x?1??e
1?x??6?lim??? x??x?1e??
1??3?x???7?lim???lim?1??x??2?xx??x?2????
??8?lim?1?x?0?x??2?x?1xxx?(x?2)?21? e?e ?1
2
?x2???e?9?lim?x???x2?1???x
?10?limsinx?sin1cosx?lim?cos1 x?1x?1x?11
1?cosxx2
?lim?0?11?limx?0x?02sinxsinx
lim(1?x)tan?x
2?12?x?1?lim(1?x)x?1?12?lim?xx?1x? cos?sin222sin?x
sin3x?2sinx3? ?13?lim2?54?sinxx?2
?14? lim
x?cotxx??
22??1
111????????? 1?21?221?2n
111111??????????????????? 2n2n1?21?2221?2211. ?????xn?
11??112. ???limn?2?2???2?? n??n??n?2?n?n???
???n?n11?n?n?1 ?n??????2?2222n??n?n??n?n??n??n?2?
2n
13. ?lim? n??n!
2n2? ???n????n!n
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发表于 2019-9-13 23:46:11