高等数学竞赛辅导材料
CH1.函数 极限与连续
1. 求y?loga?x?
1?x2的反函数.
? 2. 求y?3
x?1?x2?3x?1?x2的反函数.
?x2?2,3. 求y??2?x,
?1?x?00?x?1 的反函数.
4. 设f?x???
?1?x,?0,x?0x?0 ,求f?f?x??.
5. 设f?x?是定义在?0,
?1????内的一个函数,满足关系: f?x??f??lnx?1,求f?x?.
?x?
1
6. 设对一切实数,f?x?满足关系式: 2f?x??f??x??x2,求f?x?.
7. 设函数y?f?x?????x????的图形关于两直线x?a和x?b分别对称,证明:函数
f?x?为周期函数.
lnen?xn8. 设f?x??lim ?x?0?,试问f?x?是否连续. n??n
??
9. 求f?x??limn??
11?lnx?22n?1?的表达式.
f2?2x?x?3,求lim10. 设lim n??xf?x?n??f?4x?
2
11. 求下列极限
1) lim?x?1?tan?x 2) limnn (limx)
1xx?12 3) lim1?x?3x??82?3x
?na?nb?n5) lim??n???(a,b??2?0) ?
7) limtann(?1n??4?n)
n??x???4) limxx2?nn??(1?n?2n2) 1x?1x?1x?16)lim?a?b?c?xx?0???a?b?c???(a,b,c?0)118) limn(an?bnn???2)
3
9) 1010?1?tanx???1?sinx?lim(sinln(x?1)?sinlnx) 10) lim
x???
11) limn??sin??n2?n?
13) limn??(1?11122)(1?32)?(1?n2)
15) limn??(132n?122?22???n2)
x?0sinxlimn??(1?12)(1?1112)
22)?(1?22n)limn??(122n2n3?n3???n3)
n16) lim1n???(k?1)(k?2) k?1k4
14)
