求间断点及分类y=xcos1/x
注意备注中有点小问题,就是无穷小量是有要求的,无穷小亮是在某个过程中的,也就是该函数只有在x趋于0时是无穷小量,请明确
绛旓細闂存柇鐐箈锛0,1 lim(x~1)cos(蟺/2x)/x^2(x-1)锛-蟺/2lim(x~1)sin(蟺/2x)/(3x^2-2x)锛-蟺/2鎵浠x锛1涓哄彲鍘婚棿鏂偣 lim(x~0)cos(蟺/2x)/x^2(x-1)锛濃垶 鎵浠锛0涓烘棤绌烽棿鏂偣
绛旓細f(x)=xcos(1/x)鍦▁瓒嬭繎浜0鏃舵槸鎸崱闂存柇鐐锛氬綋x=0鏃讹紝1/x瓒嬭繎浜庢棤绌峰ぇ锛屾棤绌峰ぇ鍙彇寰堝鍊笺傛晠鏋侀檺鍦1鍜-1闂撮渿鑽°傜涓绫婚棿鏂偣锛氬彲鍘婚棿鏂偣锛堥棿鏂偣澶勫乏鍙虫瀬闄愬瓨鍦ㄤ笖鐩哥瓑锛夛紝璺宠穬闂存柇鐐癸紙闂存柇鐐瑰宸﹀彸鏋侀檺涓嶇浉绛夛級銆傜浜岀被闂存柇鐐癸細鍑℃槸闄ゅ幓涓婅堪2涓涓绫婚棿鏂偣浠ュ锛屽叏閮ㄧ殑闂存柇鐐归兘鏄浜岀被...
绛旓細褰搙瓒嬭繎涓0鏃讹紝(1-cosx)/x^2瓒嬭繎浜1/2锛堝彲鐢ㄧ瓑浠锋棤绌峰皬鏇挎崲1-cosx瓒嬭繎浜1/2x^2锛夛紝鍗硏瓒嬭繎浜0鏃讹紝f(x)涓嶇瓑浜巉(0)锛屾晠涓鸿烦璺闂存柇鐐銆
绛旓細f(x)=xcos(1/x)鍦▁瓒嬭繎浜0鏃朵笉鏄尟鑽闂存柇鐐锛屾槸鍙幓闂存柇鐐癸紒鍏跺疄浣犵湅鍒1/x棣栧厛灏卞垽鏂槸鏃犲畾涔夌偣锛屽睘浜庨棿鏂偣锛屽叾娆″氨鏄寜鐓у乏鍙虫瀬闄愮殑鍏崇郴鏉ュ垽鏂埌搴曟槸鍥涚被闂存柇鐐逛腑鐨勫摢涓绫婚棿鏂偣銆備綘璇寸殑鍑芥暟f(x)锛漻cos(1/x)鍦▁瓒嬭繎浜0鏃跺浘鍍忓涓嬶紝缁濆涓嶅睘浜庯紙娉ㄦ剰鏄笉灞炰簬锛夋尟鑽￠棿鏂偣锛屽洜涓哄嚱鏁板湪x=...
绛旓細褰搙瓒嬭繎浜0鐨勬椂鍊欙紝cos(1/x)鏄竴涓湁鐣屽嚱鏁帮紝涓攃os(1/x)涓哄懆鏈熷嚱鏁帮紝鍑芥暟鍊煎湪[-1,1]涓婂彉鍖栵紝cos(1/x)灏变細鍦-1鍒1涔嬮棿鏉ュ洖鎸崱锛屼笖x瓒婅秼浜0锛屽彉鍖栫殑瓒婂揩锛屾墍浠ユ槸绗簩绫绘尟鑽闂存柇鐐銆傝嫢鐢╩atlab鎴杕athmatic绛夋暟瀛﹁蒋浠剁敾鍑篶os(1/x)鐨勫浘鍍忥紝鍙互鐪嬪埌鍦x=0宸﹀彸鍧囨槸瀵嗛泦鐨勬尟鑽℃洸绾裤
绛旓細x=0涓哄嚱鏁扮殑鍙幓闂存柇鐐.鍥犱负璇ョ偣鍦ㄨ繖閲屾病鏈夋剰涔.绗竴绫婚棿鏂偣锛堝乏鍙虫瀬闄愰兘瀛樺湪锛夋湁浠ヤ笅涓ょ 1璺宠穬闂存柇鐐 闂存柇鐐逛袱渚у嚱鏁扮殑鏋侀檺涓嶇浉绛 2鍙幓闂存柇鐐 闂存柇鐐逛袱渚у嚱鏁扮殑鏋侀檺瀛樺湪涓旂浉绛 鍑芥暟鍦ㄨ鐐规棤鎰忎箟 绗簩绫婚棿鏂偣锛堥潪绗竴绫婚棿鏂偣锛変篃鏈変袱绉 1鎸崱闂存柇鐐 鍑芥暟鍦ㄨ鐐瑰鍦ㄦ煇涓や釜鍊兼瘮濡-1鍜+1涔嬮棿...
绛旓細f锛坸锛=cos(1\x)路cos(1\x)瀵兼暟=cos(1\x)瀵兼暟路cos(1\x)+cos(1\x)cos(1\x)瀵兼暟 =-sin(1\x)路cos(1\x)+cos(1\x)路{-sin(1\x)} =-2sin(1\x)路cos(1\x)=-sin锛2/x锛夊鏁帮紳0鏃讹紝sin锛2/x锛=0锛屸埓x=0 蟺 2蟺...k蟺 浣嗘槸x鍦ㄥ垎姣嶄綅缃紝鈭磝鈮0 鎵浠ュ湪x=0澶...
绛旓細鍥犱负鍑芥暟 y=cos²(1/x) 鍦 x=0 鐨勬瀬闄愪笉瀛樺湪锛屾墍浠ュ嚱鏁 y=cos²(1/x) 鍦 x=0 鐨闂存柇鐐圭被鍨鏄浜岀被銆
绛旓細f(x)=cos1/(x-1)鍙湁涓涓闂存柇鐐锛屽氨鏄綋X=1鏃讹紝鍑芥暟鍒嗘瘝娌℃湁鎰忔濄俧(x)=tanx/x 鏈夋棤鏁颁釜闂存柇鐐癸紝褰揦=0鏃舵垨褰揦=2K蟺+ 蟺/2 鍑芥暟鏃犳剰涔
绛旓細琛ュ厖 y=2/3 锛 x=1 x=2鏄棤绌闂存柇鐐锛岋紙鏋侀檺涓衡垶锛3.x=1鏄烦璺冮棿鏂偣锛堝乏鍙虫瀬闄愪笉鐩哥瓑锛5.1-cos(1/n)=2[sin(1/2n)]^2 鍒欏師寮=lim2[sin(1/2n)]^2/(1/n^2)鑰 n鈫掆垶鏄, sin1/n~1/n 鍒檒im2[sin(1/2n)]^2/(1/n^2)=lim2*(1/2n)^2/(1/n^2)=1/2 ...
