极限的唯一性证明 证明极限的唯一性
\u51fd\u6570\u6781\u9650\u7684\u552f\u4e00\u6027\u600e\u4e48\u8bc1\u660e\u8bbe\u5b58\u5728a\uff0cb\u4e24\u4e2a\u6570\u90fd\u662f\u51fd\u6570f(x)\u5f53x\u2192x\u3002\u7684\u6781\u9650\uff0c\u4e14a0\uff0c\u5f5300\uff0c\u5f530<\u4e28x-x\u3002\u4e28<\u03b42\u65f6\uff0c\u4f7f\u5f97\u4e28f\uff08x\uff09-b\u4e28<\u03b5\u6210\u7acb\u3002\u4e0a\u9762\u7684\u4e0d\u7b49\u5f0f\u53ef\u4ee5\u7b49\u4ef7\u53d8\u6362\u4e3aa-\u03b5<f\uff08x\uff09<a+\u03b5\u2460\u548cb-\u03b5<f\uff08x\uff09<b+\u03b5\u2461\u3002\u4ee4\u03b4=min{\u03b41\uff0c\u03b42}\uff0c\u5f530<\u4e28x-x\u3002\u4e28<\u03b4\u65f6\u3002\u2460\uff0c\u2461\u540c\u65f6\u6210\u7acb\uff0c\u5373\uff1ab-\u03b5\u2264a+\u03b5\uff0c\u79fb\u9879\u5f97\uff1a(b-a)/2\u2264\u03b5,\u56e0\u4e3a(b-a)/2\u662f\u4e00\u4e2a\u786e\u5b9a\u5927\u5c0f\u7684\u6b63\u6570\uff0c\u6240\u4ee5\u8fd9\u4e2a\u7ed3\u8bba\u4e0e\u6781\u9650\u7684\u5b9a\u4e49\uff1a\u03b5\u53ef\u4ee5\u4efb\u610f\u5c0f\u77db\u76fe\uff0c\u5047\u8bbe\u4e0d\u6210\u7acb\uff0c\u56e0\u6b64\u4e0d\u5b58\u5728a\uff0cb\u4e24\u4e2a\u6570\u90fd\u662ff\uff08x\uff09\u7684\u6781\u9650\u3002
\u6269\u5c55\u8d44\u6599
\u9996\u5148\u7528\u6781\u9650\u6982\u5ff5\u7ed9\u51fa\u2018\u5bfc\u6570\u2019\u7684\u6b63\u786e\u5b9a\u4e49\u7684\u662f\u6377\u514b\u6570\u5b66\u5bb6\u6ce2\u5c14\u67e5\u8bfa\uff0c\u4ed6\u628a\u51fd\u6570f(x)\u7684\u5bfc\u6570\u5b9a\u4e49\u4e3a\u5dee\u5546
\u7684\u6781\u9650f'(x)\uff0c\u4ed6\u5f3a\u8c03\u6307\u51faf'(x)\u4e0d\u662f\u4e24\u4e2a\u96f6\u7684\u5546\u3002\u6ce2\u5c14\u67e5\u8bfa\u7684\u601d\u60f3\u662f\u6709\u4ef7\u503c\u7684\uff0c\u4f46\u5173\u4e8e\u2018\u6781\u9650\u7684\u672c\u8d28\u2019\u4ed6\u4ecd\u672a\u63cf\u8ff0\u6e05\u695a\u3002
\u4e3a\u4e86\u6392\u9664\u6781\u9650\u6982\u5ff5\u4e2d\u7684\u76f4\u89c2\u75d5\u8ff9\uff0c\u7ef4\u5c14\u65af\u7279\u62c9\u65af\u63d0\u51fa\u4e86\u6781\u9650\u7684\u9759\u6001\u7684\u62bd\u8c61\u5b9a\u4e49\uff0c\u7ed9\u5fae\u79ef\u5206\u63d0\u4f9b\u4e86\u4e25\u683c\u7684\u7406\u8bba\u57fa\u7840\u3002\u6240\u8c13 \uff0c\u5c31\u662f\u6307\uff1a\u201c\u5982\u679c\u5bf9\u4efb\u4f55 \uff0c\u603b\u5b58\u5728\u81ea\u7136\u6570N\uff0c\u4f7f\u5f97\u5f53 n>N\u65f6\uff0c\u4e0d\u7b49\u5f0f \u6052\u6210\u7acb\u201d\u3002
\u2019\u6781\u9650\u601d\u60f3\u2019\u65b9\u6cd5\uff0c\u662f\u6570\u5b66\u5206\u6790\u4e43\u81f3\u5168\u90e8\u9ad8\u7b49\u6570\u5b66\u5fc5\u4e0d\u53ef\u5c11\u7684\u4e00\u79cd\u91cd\u8981\u65b9\u6cd5\uff0c\u4e5f\u662f\u2018\u6570\u5b66\u5206\u6790\u2019\u4e0e\u5728\u2018\u521d\u7b49\u6570\u5b66\u2019\u7684\u57fa\u7840\u4e0a\u6709\u627f\u524d\u542f\u540e\u8fde\u8d2f\u6027\u7684\u3001\u8fdb\u4e00\u6b65\u7684\u601d\u7ef4\u7684\u53d1\u5c55\u3002\u6570\u5b66\u5206\u6790\u4e4b\u6240\u4ee5\u80fd\u89e3\u51b3\u8bb8\u591a\u521d\u7b49\u6570\u5b66\u65e0\u6cd5\u89e3\u51b3\u7684\u95ee\u9898,\u6b63\u662f\u7531\u4e8e\u5176\u91c7\u7528\u4e86\u2018\u6781\u9650\u2019\u7684\u2018\u65e0\u9650\u903c\u8fd1\u2019\u7684\u601d\u60f3\u65b9\u6cd5.
\u53c2\u8003\u8d44\u6599\u767e\u5ea6\u767e\u79d1-\u6781\u9650
\u8bbe{xn}\u6781\u9650\u4e3aA\uff0c\u56de\u5fc6\u4e00\u4e0b\u6781\u9650\u5b9a\u4e49\uff0c\u4efb\u53d6\u03b5>0\uff0c\u5b58\u5728N>0\uff0c\u5f53n>N\u65f6\uff0c\u6709 |xn-A|<\u03b5
\u8bc1\u660e\u6781\u9650\u552f\u4e00\u6027\uff0c\u5047\u8bbe{xn}\u6709\u4e24\u4e2a\u6781\u9650A\uff0cB\uff0c\u4e14A>B
\u53d6\u03b5=(A-B)/2\uff0c
\u5b58\u5728N1\uff0c\u5f53n>N1\u65f6\uff0c\u6709 |xn-A|<(A-B)/2 (1)
\u5b58\u5728N2\uff0c\u5f53n>N2\u65f6\uff0c\u6709 |xn-B|<(A-B)/2 (2)
\u53d6N=max{N1\uff0cN2}\uff0c\u5219\u5f53n>N\u65f6\uff0c\u4e0a\u9762\u4e24\u5f0f\u540c\u65f6\u6210\u7acb
(1)\u53ef\u5316\u4e3a\uff1a(B-A)/2<xn-A<(A-B)/2\uff0c\u53ef\u5f97 (B+A)/2<xn<(A-B)/2+A
(2)\u53ef\u5316\u4e3a\uff1a(B-A)/2<xn-B<(A-B)/2\uff0c\u53ef\u5f97 (B-A)/2+B<xn<(A+B)/2
\u51fa\u73b0\u77db\u76fe\uff0c\u4e00\u4e2a\u5f0f\u5b50\u662fxn>(A+B)/2\uff0c\u53e6\u4e00\u4e2a\u662fxn<(A+B)/2
\u56e0\u6b64\u6781\u9650\u552f\u4e00\u3002
当n→∞时若xn→a,且xn→b,a>b.
取ε=(a-b)/2,存在正整数N,使得当n>N时|xn-a|<ε,
∴a-ε<xn<a+ε,①
同理b-ε<xn<b+ε,②
a-ε=(a+b)/2=b+ε,
①②矛盾。
∴数列的极限是唯一的。
简单计算一下即可，答案如图所示
绛旓細鍗筹細b-蔚鈮+蔚锛岀Щ椤瑰緱锛(b-a)/2鈮の,鍥犱负(b-a)/2鏄竴涓‘瀹氬ぇ灏忕殑姝ｆ暟锛屾墍浠ヨ繖涓粨璁轰笌鏋侀檺鐨瀹氫箟锛毼靛彲浠ヤ换鎰忓皬鐭涚浘锛屾墍浠ュ亣璁句笉鎴愮珛锛屽洜姝や笉瀛樺湪a锛宐涓や釜鏁伴兘鏄痜锛坸锛夌殑鏋侀檺锛岄櫎闈瀉=b鐭涚浘鎵嶄笉浼氬嚭鐜般傚樿嫢鏄痻瓒嬩簬鏃犵┓澶ф椂鐨勫敮涓鎬ц瘉鏄鍙互鍙傜湅楂樻暟涔︽暟鍒楁瀬闄愬敮涓鎬ц瘉鏄庯紝璇佹硶瀹屽叏涓鏍...
绛旓細鍦ㄨ瘉鏄庢瀬闄愬瓨鍦ㄤ笌鍚︾殑闂涓紝鏋侀檺鐨勫敮涓鎬у彲浠ョ洿鎺ヤ娇鐢紝涓嶅繀鍔犱互璇佹槑锛屾瀬闄愮殑鍞竴鎬у彲浣滀负瀹氱悊浣跨敤锛岃屽钩鏃惰瘉鏄庨鏃讹紝瀹氱悊鏄笉鐢ㄥ姞浠ヨ瘉鏄庣殑; 鏀舵暃鏁板垪鐨鏋侀檺鐨勫敮涓鎬ц瘉鏄濡備笅:璇佹槑锛氬亣璁炬暟鍒梐n鏀舵暃浜庡疄鏁癆鍜屽疄鏁癇锛屽叾涓瑼鈮燘锛屼笉濡ㄥ亣璁続<B銆傞偅涔堝浜庝换缁欑殑蔚锛屾诲瓨鍦∟>0锛屼娇寰楀浜庝换鎰忕殑n鈮锛屾绘湁 ...
绛旓細鍋囧鏋侀檺涓嶅敮涓锛岃lim f(x)=a,lim f(x)=b,涓嶅Θ璁綼0,褰0<|x-x0|<未鏃讹紝鏈 |f(X)-a|<蔚, f(X)-b||<蔚,涔熷氨鏄 f(x)b-蔚=(a+b)/2,鐭涚浘銆傛墍浠鏋侀檺鍞竴
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绛旓細鍞竴锛屽鏋滀笉鍞竴灏变笉鑳界О涔嬩负鏋侀檺銆備竴鑸潵璇达紝鍙湁鏌愪竴鐗瑰畾鐨勭偣浼氬嚭鐜板乏鍙虫瀬闄愪笉涓鑷寸殑鎯呭喌锛岄偅鏍锋瀬闄愬氨涓嶅瓨鍦紝涔熷彲浠ョ悊瑙ｄ负鏋侀檺涓嶅敮涓銆傛墍璋撲笉鍞竴锛屼笉鏄鏈変袱涓紝鑰屾槸璇磋繛涓涓兘娌℃湁銆鍞竴鎬鏄ц川锛屽氨鏄鍙鏋侀檺瀛樺湪锛屽氨鏄敮涓鐨勶紝涓嶅彲浠ヨ兘鏈夊涓殑锛屽鏋滄湁宸﹀彸鏋侀檺锛屼笉涓鑷达紝閭ｄ篃涓嶆槸鏈変袱...
绛旓細鍏堢湅鍞竴鎬濡備綍璇佹槑鍑烘潵锛氳锛歭im an=a锛屽悓鏃讹紝lim an=b锛屽垯璇侊細a=b 鏍规嵁瀹氫箟锛氫换鎰徫>0锛屽瓨鍦∟1>0锛屽綋n>N1锛屾湁|an-a|<蔚/2 瀵逛笂杩拔>0锛屽瓨鍦∟2>0锛屽綋n>N2锛屾湁|an-b|<蔚/2 鍙朜=max{N1,N2}锛屽垯褰搉>N锛屼笂涓や笉绛夊紡閮芥垚绔 浜庢槸锛寍a-b|鈮an-a|+|an-b|<蔚/2+蔚/2=蔚...
绛旓細璇佹槑锛氬亣璁炬暟鍒梐n鏀舵暃浜庡疄鏁癆鍜屽疄鏁癇锛屽叾涓瑼鈮燘锛屼笉濡ㄥ亣璁続<B銆傞偅涔堝浜庝换缁欑殑e锛屾诲瓨鍦∟>0锛屼娇寰楀浜庝换鎰忕殑n鈮锛屾绘湁 |an-A|<e 鍙杄=(B-A)/2锛岄偅涔堝浜庝换鎰忕殑n鈮锛屽繀鏈 |an-A|<(B-A)/2 鍗矨-(B-A)/2<an<A+(B-A)/2 鍗(3A-B)/2<an<(A+B)/2 鍥犳 (3A-B)/2-B...
绛旓細鍋囪f(x)鍦▁0鐐瑰鏈変袱涓鏋侀檺A>B 閭ｄ箞鐢辨瀬闄愬畾涔夛紝鍙杄psilon=(A-B)/3>0 瀛樺湪delta>0,浣垮緱褰搢x-x0|<delta鏃讹紝鏈 f(x)>A-epsilon锛屽苟涓攆(x)<B+epsilon 鐢变簬A-epsilon>B+epsilon,鎵浠ヨ繖鏄煕鐩剧殑
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