数学分析试题集锦（一）：05

??c±s?′? ?#?¨ x1±s?′? ? 1
￡ ü

1
Z?x3 ?3x + c = 0
c ^è ?
uW[0;1]=? V? μ ??]￥ L?

2
Z?xn +px+q = 0
n1?? ?
p;q1 L ?
?n1} ? Hà μ ? L?
?n1 ? Hàμ?? L?b 2
 !f(x) = xm(1?x)n;m;n1?? ?
x 2 [0;1]
5i? 2 (0;1)
 P m n = ? 1?? 3
?¨ ?ì μ°?′? ?￡ ü/ ?? T

1
jsinx?sinyj6jx?yj;x;y 2 (?1;+1);
2
jxj6jtanxj;x 2 (?…2; …2);?|? ??O??x = 0

3
ex > 1+x;x 6= 0;
4
y?xy < ln yx < y?xx ;) < x < y;
5
x1+x2 < arctanx < x;x > 0. 4
 !f ??a μ ??￥=¨? ?
￡ ü lim h!0 f(a+h)+f(a?h)?2f(h) h2 = f 00(a): 5
 !limx!+1f0(x) = a
p￡
?iT > 0
μ limx!+1[f(x+T)?f(x)] = Ta 6.f ?f(x)[a;b] V?
?a > 0
￡ ü
i? 2 (a;b)
 P¤ 2?[f(b)?f(a)] = (b2 ?a2)f0(?): 1 7
 !f(x)(a;+1)  V?
Olim x!a+ f(x) = limx!+1f(x) = A ,p￡
i ?(a;+1)
 Pf0(?) = 0b 8
 !f(x) V?
p￡
f(x)  ,?-WB?μf(x)+f0(x)￥ ,?. 9
 !f ?f(x)x0?í ??
"x0?? V?
Olimx!x 0 f0(x) = A
 p￡
f0(x)i
Of0(x) = A . 10
?f(x)(a;b) V?
Of0(a) 6= f0(b)
k1o?f0(a)?f0(b)-W ￥?B L ?
5à iB?? 2 (a;b)
 Pf0(?) = k . 11
 !f ?f(x)(a;b)= V?
Of0(x)??
￡ üf0(x)(a;b) ? ?. 12
?f ?f(x);g(x)?h(x)[a;b] ??
(a;b) V?
￡ üi? 2 (a;b)
 P¤fl flfl flfl flfl flfl f(a) g(a) h(a) f(b) g(b) h(b) f0(?) g0(?) h0(?) flfl flfl flfl flfl fl = 0 V??2T? ? μ°?′? ?? O?′? ?b 13
 !f(x)(1;+1) ??
Olimx!§1 = +1
￡ ü
f(x)(1;+1) |? ?￥Kl′. 14
 !f(x)[a;b) ??
lim x!b? f(x) = B .
1
?ix 2 [a;b)
 Pf(x1) > B
5f(x)[a;b) r?Kv′

2
?Tix 2 [a;b)
 Pf(x1) = B
??yf(x)[a;b) r? Kv′
$ 15
 !f(x)[a;+1)μ?
f0(x)i
Olimx!+1f0(x) = b .p￡b = 0 . 16
p￡
arctanx+arccosx = …2(jxj6 1) . 2 x2±s?′? ?#?¨ 1
p/ ???￥K

1
limx!0 tanaxsinbx ;
2
limx!0 1?cosx2x3 sinx ;
3
limx!0 ln(1+x)?xcosx?1 ;
4
limx!0 tanx?xx?sinx;
5
limx!0(1x ? 1ex?1);
6
limx!0 lncosaxlnsinbx;
7
lim x!…2 tanx?6 secx+5;
8
limx!1( 1lnx ? 1x?1);
9
limx!…(… ?x)tan x2;
10
limx!1x 11?x;
11
limx!+1 xbeax (a;b > 0);
12
limx!+1 …2?arctanxsin 1 x ;
13
limx!+1 lnc xxb (b;c > 0);
14
lim x!0+ lnc x xb (b;c > 0);
15
lim x!…6 1?2sinx cos3x ;
16
lim x!0+ lnx cotx;
17
limx!0 (1+x) 1x?e x ; 3
18
lim x!0+ xsinx;
19
lim x!0+ (ln 1x)x;
20
limx!0(tanxx ) 1x2 ;
21
limx!0( 1x2 ? 1sin2 x);
22
lim x!0+ sinxlnx. 2
f ?f(x)[0;x] ?¨ ?ì μ°?′? ?μ f(x)?f(0) = f0( x)x; 2 (0;1): k￡/ f ?μlim x!0+ = 12

1
f(x) = ln(1+x);
2
f(x) = ex. 3
 !f(x)=¨ V?
p￡
 lim h!0 f(x+2h)?2f(x+h)+f(x) h2 = f 00(x): 4
/ f ???¨ ?ArE5pK

1
limx!0 x2 sin 1xsinx ;
2
limx!1 x+sinxx?cosx;
3
limx!1 2x+sin2x(2x+sinx)esinx;
4
limx!1 (x2?1)sinxln(1+sin … 2x) . x3f ?￥ 6?aj??f ?Tm 1
?¨f ?￥???￡ ü/ ?? T
 4
1
2…x < sinx < x;x 2 (0; …2);
2
x < sinx < x? x36 ;x < 0;
3
x? x22 < ln(1+x) < x;x > 0;
4
tanx > x+ x33 ;x 2 (0; …2);
5
2px > 3? 1x;x < 0: 2
??/ f ?￥??uW

1
y = x3 ?6x;
2
y = p2x?x2;
3
y = 2x2 ?lnx;
4
y = x2?1x ;
5
y = 2x2 ?sinx;
6
y = xne?x; 3
p/ f ?￥′

1
y = x?ln(1+x);
2
y =;x+ 1x;
3
y = 1+3xp4+5x2;
4
y = (lnx)2x ;
5
y = 2x3 ?x4;
6
y = arctanx? 12 ln(1+xx); 5 4
 !f(x) = 8 < : x4 sin2 1x; x 6= 0; 0; x = 0:
1
￡ ü
x = 0 ^f ?￥l′?

2
 a üf￥l′?x = 0) ^? ?@′￥?B sHq? = sHq. 5
￡ ü
?f ?f(x)?x0)μf0+(x0) < 0;f0?(x0) > 0
5x01f ￥v′?. 6
 !f(x) = alnx + bx2 + xx1 = 1;x2 = 2)?|￥′
 k?a ?b￥′
iù? Hfx1?x2 ^|¤v′? ^l′
7.(1)pf ?f(x) = ax?lnxx > 0 ￥′
(2)pZ?ax = lnxμ ?? L?￥Hq. 8. !f(x)
g(x) Là  ?? V±
O flfl flfl flfl f(x) g(x) f0(x) g0(x) flfl flfl flfl > 0 p￡
f(x)￥  L?-WB?μg(x) = 0￥?. 9.??/ f ?￥j?uWD.?

1
y = 3x2 ?x3;
2
y = x2 + 1x;
3
y = ln(1+x2);
4
y = p1+x2; 10.￡ üwLy = x+1x2+1μê?]B°L ￥??.?. 6 11.ùa;b1?′ H
?(1;3)1wLy = ax3 +bx2￥.?
$ 12.￡ ü
 (1)?f(x)1/jf ?
?1dμ L ?
5?f(x)1/jf ?
(2)?f(x)ag(x) (1/jf ?
5f(x)+g(x)1/jf ?
(3)?f(x)1uWI ￥/jf ?
g(x)1J ￥/j?9f ?
f(I) ‰ J
5g –f(x)1I ￥/jf ?. 13. !f(x)1uWI ?ì jf ?
￡ ü
?x0 2 I1f(x)￥l′ ?
5x01f(x)I ·B￥l′?. 14.?¨/jf ?àQ￡ ü?/?? T
 (1)?i L ?a;b;μ ea+b2 612(ea+eb): (2)??dμf ?a;b;μ 2arctan a+b2 > arctana+arctanb: 15.??ê4? ?h > 0
Z? PwL y = hp…e?h2x2 x = § ( > 0)1ó?￥è ?
)μ.?. 16.py = x2x2+1￥′#.?
ip.?)￥MLZ?. 17.T/ f ?￥m?

1
y = x3 ?6x;
2
y = e?(x?1)2; 7
3
y = 1x2?1;
4
y = ln 1+x1?x;
5
y = x?2arctanx;
6
y = xe?x;
7
y = x2?2x?3x2+1 ;
8
y = (x?1)3(x+1)3;
9
y = x4(1+x)3. x4f ?￥Kv′Kl′ù5 1
p/ f ?·?uW ￥Kv′DKl′
1
y = x5 ?5x4 +5x3 +1;
2
y = 2tanx?tan2 x;x 2 (0; …2);
3
y = pxlnx;
4
y = jx2 ?3x+2j;
5
y = ejx?3j;[?5;5]; 2
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 kü ?s? 
 P[?  1H ???￥ ? ?1Kv. 3. !¨Né??  H
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?? P ?D?n? ?-μ￥üZ?1K l. 4.p=¤???x2a2 + y2b2 = 17Hü??USà￥ ?Kv￥ ?. 8 5.?M(p;q)??tLy2 = 2pxK  ?. 6.SB????Q V
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?HbD|úl1é H A ?Kl.
? ü
 A?Kl H{?V ￡? ?Kv.
 8. !? g￥?1fi
??￥ ?1v0m=s
? g|Te?
?? H W|Tt = 0
?9 bE ? H
??￥?Z?1
 8 < : x = tv0 cosfi y = tv0 sinfi? 12gt2 ? ?v0?M
ù????? g￥?fi
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