2所以当x?3时y?x?9为无穷小?
x?3 (2)当x?0时|y|?|x||sin1|?|x?0|? 因为???0? ???? ? 当0?|x?0|??时? 有
x|y|?|x||sin1|?|x?0|?????
x所以当x?0时y?xsin1为无穷小?
x
3? 根据定义证明? 函数y?1?2x为当x?0时的无穷大? 问x应满足什么条件?
x能使|y|?104？
证明 分析|y|?1?2x?2?1?1?2? 要使|y|?M? 只须1?2?M? 即
xx|x||x||x|?1?
M?21? 使当0?|x?0|??时? 有1?2x?M?
xM?2所以当x?0时? 函数y?1?2x是无穷大?
x1? 当0?|x?0|?1时? |y|?104? 取M?104? 则??410?2104?2 4? 求下列极限并说明理由? (1)lim2x?1;
x??x21?x (2)lim? x?01?x 解 (1)因为2x?1?2?1? 而当x?? 时1是无穷小? 所以lim2x?1?2?
x??xxxx221?x1?x (2)因为?1?x(x?1)? 而当x?0时x为无穷小? 所以lim?1?
x?01?x1?x 5? 根据函数极限或无穷大定义? 填写下表? 证明 因为?M?0? ??? x?x0 f(x)?A ???0? ???0? 使 当0?|x?x0|??时? 有恒|f(x)?A|??? f(x)?? f(x)??? f(x)??? x?x0? x?x0? x?? ???0? ?X?0? 使当|x|?X时? 有恒|f(x)|?M? x??? x??? 解 x?x0 f(x)?A ???0? ???0? 使当0?|x?x0|??时? 有恒|f(x)?A|??? ???0? ???0? 使当0?x?x0??时? 有恒|f(x)?A|??? ???0? ???0? 使当0?x0?x??时? 有恒|f(x)?A|??? ???0? ?X?0? 使当|x|?X时? 有恒|f(x)?A|??? ???0? ?X?0? 使当x?X时? 有恒|f(x)?A|??? ???0? ?X?0? 使当x??X时? 有恒|f(x)?A|??? f(x)?? ?M?0? ???0? 使当0?|x?x0|??时? 有恒|f(x)|?M? ?M?0? ???0? 使当0?x?x0??时? 有恒|f(x)|?M? ?M?0? ???0? 使当0?x0?x??时? 有恒|f(x)|?M? ???0? ?X?0? 使当|x|?X时? 有恒|f(x)|?M? ???0? ?X?0? 使当x?X时? 有恒|f(x)|?M? ???0? ?X?0? 使当x??X时? 有恒|f(x)|?M? f(x)??? ?M?0? ???0? 使当0?|x?x0|??时? 有恒f(x)?M? ?M?0? ???0? 使当0?x?x0??时? 有恒f(x)?M? ?M?0? ???0? 使当0?x0?x??时? 有恒f(x)?M? ???0? ?X?0? 使当|x|?X时? 有恒f(x)?M? ???0? ?X?0? 使当x?X时? 有恒f(x)?M? ???0? ?X?0? 使当x??X时? 有恒f(x)?M? f(x)??? ?M?0? ???0? 使当0?|x?x0|??时? 有恒f(x)??M? ?M?0? ???0? 使当0?x?x0??时? 有恒f(x)??M? ?M?0? ???0? 使当0?x0?x??时? 有恒f(x)??M? ???0? ?X?0? 使当|x|?X时? 有恒f(x)??M? ???0? ?X?0? 使当x?X时? 有恒f(x)??M? ???0? ?X?0? 使当x??X时? 有恒f(x)??M? x?x0? x?x0? x?? x??? x??? 6? 函数y?xcos x在(??? ??)内是否有界？这个函数是否为当x??? 时的无穷大？为什么？
解 函数y?xcos x在(??? ??)内无界?
这是因为?M?0? 在(??? ??)内总能找到这样的x? 使得|y(x)|?M? 例如
y(2k?)?2k? cos2k??2k? (k?0? 1? 2? ? ? ?)?
当k充分大时? 就有| y(2k?)|?M?
当x??? 时? 函数y?xcos x不是无穷大?
这是因为?M?0? 找不到这样一个时刻N? 使对一切大于N的x? 都有|y(x)|?M? 例如
y(2k???)?(2k???)cos(2k???)?0(k?0? 1? 2? ? ? ?)?
222对任何大的N? 当k充分大时? 总有x?2k????N? 但|y(x)|?0?M?
2 7? 证明? 函数y?1sin1在区间(0? 1]上无界? 但这函数不是当x?0+时的无穷
xx大?
证明 函数y?1sin1在区间(0? 1]上无界? 这是因为
xx ?M?0? 在(0? 1]中总可以找到点xk? 使y(xk)?M? 例如当
xk?1(k?0? 1? 2? ? ? ?)
?2k??2时? 有
y(xk)?2k????
2当k充分大时? y(xk)?M?
当x?0+ 时? 函数y?1sin1不是无穷大? 这是因为
xx ?M?0? 对所有的??0? 总可以找到这样的点xk? 使0?xk??? 但y(xk)?M? 例如可取
xk?1(k?0? 1? 2? ? ? ?)?
2k?当k充分大时? xk??? 但y(xk)?2k?sin2k??0?M?
习题1?5
1? 计算下列极限?
2x (1)lim?5? x?2x?322x?52??5??9? 解 limx?2x?32?32x (2)lim2?3? x?3x?12(3)2?3x?3 解 lim2??0? x?3x?1(3)2?12x2x?1? (3)lim?x?1x2?12(x?1)2x?2x?1x?1?0?0?lim?lim 解 lim? x?1x?1(x?1)(x?1)x?1x?12x2?1324x?2x?x? (4)limx?03x2?2x3224x?2x?x4x?2x?1?1? 解 lim?limx?03x2?2xx?03x?22(x?h)2?x2 (5)lim?
h?0h222(x?h)2?x2x?2hx?h?x 解 lim?lim?lim(2x?h)?2x? h?0h?0h?0hh (6)lim(2?1?1)?
x??xx21?lim1?2? 解 lim(2?1?1)?2?limx??x??xx??x2xx22x (7)lim2?1? x??2x?x?11?122?1?lim1? x? 解 limxx??2x2?x?1x??2?1?122xx2xx? (8)lim4?2x??x?3x?12x?0(分子次数低于分母次数? 极限为零)? 解 lim4x?2x??x?3x?11?12x?limx2x3?0? 或 lim4x?2x??x?3x?1x??11?2?x2x42 (9)limx2?6x?8?
x?4x?5x?42x?6x?8?lim(x?2)(x?4)?limx?2?4?2?2?
解 lim2x?4x?5x?4x?4(x?1)(x?4)x?4x?14?13 (10)lim(1?1)(2?1)? 2x??xx1)?lim(2?1)?1?2?2? 解 lim(1?1)(2?1)?lim(1?x??xx2x??xx??x2 (11)lim(1?1?1? ? ? ? ?1)?
n??242n1?(1)n?12 解 lim(1?1?1? ? ? ? ?1)?lim?2?
n??242nn??1?12 (12)lim1?2?3? ? ? ? ?(n?1)?
n??n2(n?1)n1?2?3? ? ? ? ?(n?1)2?1limn?1?1? ?lim 解 limn??n??2n??n2n2n2(n?1)(n?2)(n?3)?
n??5n3(n?1)(n?2)(n?3)1? (分子与分母的次数相同? 极限为 解 limn??55n3最高次项系数之比)?
(n?1)(n?2)(n?3)11)(1?2)(1?3)?1? 或 lim?lim(1?n??5n??nnn55n3 (14)lim(1?33)?
x?11?x1?x (13)lim2131?x?x?3??lim(1?x)(x?2) ?)?lim 解 lim(x?11?x1?x3x?1(1?x)(1?x?x2)x?1(1?x)(1?x?x2)x?2??1? x?11?x?x2 2? 计算下列极限? ??lim32x?2x (1)lim? x?2(x?2)232(x?2)20x?2x??? 解 因为lim3??0? 所以limx?2(x?2)2x?2x?2x2162x (2)lim? x??2x?12x 解 lim?? (因为分子次数高于分母次数)? x??2x?1 (3)lim(2x3?x?1)?
x?? 解 lim(2x3?x?1)??(因为分子次数高于分母次数)?
x?? 3? 计算下列极限?
同济大学第六版高等数学课后答案详解全集
