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第一章 函数与极限
习题1.1 A组
1.(1)定义域为{x|x?k?}(k?Z) (2){x|x??1}. 2.f(x+1)=(x+1)?(3x?1)?2?x?x f()=()?221x1x2313?2?2??2 xxx222 f(x?x)?f(x)?(x?x)?3(x?x)?2?(x?3x?2)?x?2xx?3x 3.f(?4)?1?(?4)2?17,f(0)?sin0?0,f(2)?sin2,f(?)?sin??0. 4.(1)y=sin (2)y=e211 是由y?u2,u?sinv,v?复合而成。xx是由y=eu,u?sinv,v?2x?1复合而成。
sin(2x?1) 5.(1)f[?(x)]?arcsin(lgx),1?x?10. 10 (2)f[?(x)]?1x1?1x2?x,x?0. 21?x1,又对?x?0,易见f(x)?0. 36. 对?x?0,2?0,从而f(x)?1x1故0?f(x)?,对?x?0,从而f(x)在定义域上有界.
3 7.圆锥形漏斗的底面半径r?R?RR?,高h?R2?()2?2?2?2?224?2??2 121R?2R)?所以V=?r?h??(332?2? 8.p?? B组
1R3?2224????4??? 224?3w,0?w?10?
?30?2.7(w?10),w?101.(1)0?sinx?1,?2k??x?2k???.(k?Z)
(2){
0?x?a?1
0?x?a?1
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1时,定义域为?. 21当0?a?时,定义域为?a,1?a?。
2所以当a>2. f?1(x)???lnx,0?x?1 2x?1,x?1?x2x1?x3.f(f(x))?f2(x)? ?2x1?2x1?()21?x2x2x1?x f3(x)? ?2x1?3x1?2()21?x2 ?fn(x)?x1?nx2
4.(1)y=arctanu,u=v+w,v=e,w?sinz,z?x (2)y=2,u?arcsinv,v?ux21. 21?x5.(1)令x=-1,则f(1)?f(?1)?f(2)
?f(2)?a?(?a)?2a.f(n?2)?f(n)?f(2)?f(n)?f(n?2)?f(n)?na.
(2)f(x)以2为周期的周期函数?a?0. 习题1.2
1.(1)无极限 (2)limxn?0; (3)无极限.
n??2. (1)?;(2)?.
3.证明:?M?0,?n都有|xn|?M
???0,?N?0,?n?N,都有|yn|??,取?1? ???0,|xnyn?0|?|xn||yn|?M? ?limxnyn?0.
n???M,有|yn|??M
?M??
4.
R??limx2k?1?a,limx2k?a,则对于???0,总存在N1和N2,当k?N1时
R??实用文档
|x2k?1?a|??
|x2k?a|?? 当k>N2时, 取N?max{2N1?1,2N2}.当k?N时,不论n为奇数或是偶数,总有 |xn?a|?? ?limxn?a.
n?? 5.
limun?a,?对???0,?N?0,当n?N时,恒有|un?a|??成立.
n?? 由于||un?|a||?|un?a|??,?lim|un|?|a|
n?? 逆命题不成立,如xn?(?1),lim|xn|?1,但limxn不存在
n??n??n
习题1.3 A组
1.(1)对???0,????3?0,当0?|x?3|??时,有
|3x?3?6|?|3x?9|?3|x?3|?3? ?lim(3x?3)?6
x?3?3??
(2)对???0,????2,当0<|x-1|
?lim(2x?8)?10
x?1 (3) 对???0,????,当0<|x-2|
x2?4?4|?|x?2|?? 有|x?2x2?4?4 ?limx?2x?2 (4)
11-4x21????0,???,当0<|x+|
222x?122?