高等数学教案1－5

§1．4 函数的极限 §1( 6 极限运算法则 定理1 有限个无穷小的和也是无穷小( 例如( 当x(0时( x与sin x都是无穷小( x(sin x也是无穷小( 简要证明( 设(及(是当x(x0时的两个无穷小( 则(( (0( ((1(0及(2(0( 使当0(|x(x0|((1时( 有|(|(( ( 当0(|x(x0|((2时( 有|(|(( ( 取( (min{(1( (2}( 则当0(|x(x0|((时( 有|(((|(|(|(|(|(2( ( 这说明(((也是无穷小( 证明( 考虑两个无穷小的和( 设(及( 是当x(x0时的两个无穷小( 而( (((( ( 任意给定的( (0( 因为( 是当x(x0时的无穷小( 对于 (0存在着(1(0( 当0(|x(x0|((1时( 不等式 |(|( 成立( 因为(是当x(x0时的无穷小( 对于 (0存在着(2(0( 当0(|x(x0|((2时( 不等式 |(|( 成立( 取( (min{(1( (2}( 则当0(|x(x0|((时( |(|( 及|(|( 同时成立( 从而|(|(|(((|(|(|(|(|( ( (( ( 这就证时了( 也是当x(x0时的无穷小( 定理2 有界函数与无穷小的乘积是无穷小( 简要证明( 设函数u在x0的某一去心邻域{x|0(|x(x0|((1}内有界( 即(M(0( 使当0(|x(x0|((1时( 有|u|(M( 又设( 是当x(x0时的无穷小( 即(( (0( 存在(2 (0( 使当0(|x(x0|((时( 有|(|(( ( 取( (min{(1( (2}( 则当0(|x(x0|((时(有 |u((|( M( ( 这说明u((也是无穷小( 例如( 当x((时( 是无穷小( arctan x是有界函数( 所以 arctan x也是无穷小( 推论1 常数与无穷小的乘积是无穷小( 推论2 有限个无穷小的乘积也是无穷小( 定理3 如果lim f (x)(A( lim g (x)(B( 那么 (1) lim [f (x)(g(x)] ( lim f (x) (lim g (x) (A ( B ( (2) lim f (x)(g(x) ( lim f (x) ( lim g (x) (A(B ( (3) (B(0)( 证明(1)( 因为lim f (x)(A( lim g (x)(B ( 根据极限与无穷小的关系( 有 f (x)(A((( g (x)(B((( 其中(及( 为无穷小( 于是 f (x) ( g (x)((A (() ( (B (() ((A ( B) (((( ()( 即f (x) ( g (x)可表示为常数(A ( B)与无穷小((( ()之和( 因此 lim [f (x) ( g (x)] (lim f (x) ( lim g (x) (A ( B ( 推论1 如果lim f (x)存在( 而c为常数( 则 lim [c f (x)](c lim f (x)( 推论2 如果lim f (x)存在( 而n是正整数( 则 lim [f (x)]n ([lim f (x)]n( 定理4 设有数列{xn }和{yn }( 如果 ( ( 那么 (1) ( (2) ( (3)当 (n(1( 2( ( ( ()且B(0时( ( 定理5 如果(x)((x)( 而lim (x)(a ( lim ((x)(b ( 那么a(b ( 例1( 求 ( 解( ( 讨论( 若 ( 则 提示( (a0x0n(a1x0n(1(( ( ((an(P(x0)( 若 ( 则 ( 例2( 求 ( 解( EMBED Equation.3 EMBED Equation.3 ( 提问( 如下写法是否正确？ EMBED Equation.3 ( EMBED Equation.3 ( 例3( 求 ( 解( ( 例4( 求 ( 解( ( 根据无穷大与无穷小的关系得 ((( 提问( 如下写法是否正确？ ( 讨论( 有理函数的极限 提示 当 时( ( 当 且 时( ( 当Q(x0)(P(x0)(0时( 先将分子分母的公因式(x(x0)约去( 例5 求 ( 解( 先用x3 去除分子及分母( 然后取极限( ( 例6 求 ( 解( 先用x3 去除分子及分母( 然后取极限( ( 例7( 求 ( 解( 因为 ( 所以 ( 讨论( 有理函数的极限 提示  例8( 求 ( 解( 当x((时( 分子及分母的极限都不存在( 故关于商的极限的运算法则不能应用( 因为 ( 是无穷小与有界函数的乘积( 所以 ( 定理8(复合函数的极限运算法则) 设函数y(f[g(x)]是由函数y(f(u)与函数u(g(x)复合而成( f[g(x)]在点x0的某去心邻域内有定义( 若 ( ( 且在x0的某去心邻域内g(x)(u 0( 则 ( 定理8(复合函数的极限运算法则) 设函数y(f[g(x)]是由函数y(f(u)与函数u(g(x)复合而成( f[g(x)]在点x0的某去心邻域内有定义( 若g(x)(u0(x(x0)( f(u)(A(u(u0)( 且在x0的某去心邻域内g(x)(u0( 则 ( 简要证明 设在{x|0(|x(x0|((0}内g(x)(u0( 要证(( (0( (((0( 当0(|x(x0|(( 时( 有|f[g(x)](A|(( ( 因为f(u)(A(u(u0)( 所以(( (0( (((0( 当0(|u(u0|((时( 有|f(u)(A|(( ( 又g(x)(u0(x(x0)( 所以对上述((0( ((1(0( 当0(|x(x0|((1时( 有|g(x)(u0|((( 取((min{(0( (1}( 则当0(|x(x0|((时( 0<|g(x)(u0|((( 从而 |f[g(x)](A|(|f(u)(A|(( ( 注( 把定理中 换成 或 ( 而把 换成 可类似结果( 把定理中g(x)(u0(x(x0)换成g(x)(((x(x0)或g(x)(((x(()( 而把f(u)(A(u(u0)换成f(u)(A(u(()可类似结果( 例如 例9 求 ( 解 是由 与 复合而成的( 因为 ( 所以 ( 5 _1096091625.unknown _1096091642.unknown _1096091651.unknown _1096091655.unknown _1096091660.unknown _1096091662.unknown _1096092570.unknown _1096092822.unknown _1096491273.unknown _1096092605.unknown _1096092566.unknown _1096092483.unknown _1096091661.unknown _1096091658.unknown _1096091659.unknown _1096091656.unknown _1096091653.unknown _1096091654.unknown _1096091652.unknown _1096091647.unknown _1096091649.unknown _1096091650.unknown _1096091648.unknown _1096091645.unknown _1096091646.unknown _1096091643.unknown _1096091634.unknown _1096091638.unknown _1096091640.unknown _1096091641.unknown _1096091639.unknown _1096091636.unknown _1096091637.unknown _1096091635.unknown _1096091630.unknown _1096091632.unknown _1096091633.unknown _1096091631.unknown _1096091628.unknown _1096091629.unknown _1096091626.unknown _1096091607.unknown _1096091617.unknown _1096091621.unknown _1096091623.unknown _1096091624.unknown _1096091622.unknown _1096091619.unknown _1096091620.unknown _1096091618.unknown _1096091612.unknown _1096091615.unknown _1096091616.unknown _1096091614.unknown _1096091609.unknown _1096091611.unknown _1096091608.unknown _1096091599.unknown _1096091603.unknown _1096091605.unknown _1096091606.unknown _1096091604.unknown _1096091601.unknown _1096091602.unknown _1096091600.unknown _1096091595.unknown _1096091597.unknown _1096091598.unknown _1096091596.unknown _1096091592.unknown _1096091593.unknown _1096091591.unknown _1085380463.unknown