x*cos（1/x）的极限怎么求？
\u6c42(cos\u03c0x/2)/(x^2-1)(x\u21921)\u7684\u6781\u9650 \u6c42\u8be6\u7ec6\u8fc7\u7a0b
\u5e0c\u671b\u5bf9\u60a8\u6709\u5e2e\u52a9\u54c8
lim(x->0)(cos(x)-1)/x
=lim(x->0)-2(sin(x/2))^2/x
\u7528\u7b49\u4ef7\u65e0\u7a77\u5c0f\u7684\u66ff\u6362,\u5373\u7528x/2\u66ff\u6362sin(x/2)
\u5219\u539f\u5f0f=lim(x->0)-x/2=0
绛旓細褰 x 瓒嬭繎浜 0 鏃讹紝鍑芥暟 f(x) = 1/x*cos(1/x^2) 鐨勫兼尟鑽℃棤绌峰娆★紝涓旀尟鑽￠鐜囪秺鏉ヨ秺蹇紝鍥犳鍦 x = 0 澶勬棤娉曞彇寰楁瀬闄愬笺傚洜姝わ紝鍑芥暟 f(x) 鍦 x = 0 澶勬槸涓涓尟鑽￠棿鏂偣銆傝繖鏄洜涓哄湪 x 瓒嬭繎浜 0 鏃讹紝cos(1/x^2) 鐨勬尟鑽￠鐜囪秺鏉ヨ秺楂橈紝鑰 1/x 鐨勭粷瀵瑰间篃瓒婃潵瓒婂ぇ锛屽鑷...
绛旓細鏈変袱绉嶅仛娉曪細1.鐩存帴绠楀嚭鏉 lim(x鈫0) xcos(1/x)=0 鍥犱负x鏄棤绌峰皬閲,cos(1/x)鏄湁鐣岄噺,鏃犵┓灏忛噺涔樹互鏈夌晫閲忎负鏃犵┓灏忛噺 2.鍒╃敤瀹氫箟鏉ヨ瘉鏄 鑰冭檻 |xcos(1/x)-0| =|x| * |cos(1/x)| 鈮x| * 2 =2|x| 鍥犳,瀵逛簬浠绘剰蔚>0,鍙栁=蔚/2>0,瀵逛换鎰弜 ...
绛旓細x_degrees = x * (180 / pi)杩欓噷鐨 pi 鏄渾鍛ㄧ巼锛岀害绛変簬 3.14159銆傞渶瑕佹敞鎰忕殑鏄紝浣欏鸡鍑芥暟鏄懆鏈熸х殑锛屽嵆 cos(x) = cos(x + 2蟺n)锛屽叾涓 n 鏄换鎰忔暣鏁般傚洜姝わ紝鍦ㄨВ鏂圭▼鏃讹紝鍙兘鏈夊涓В銆備綘鍙互鍦ㄧ粰瀹氱殑鐗瑰畾鑼冨洿鍐呴夋嫨涓涓В浣滀负缁撴灉锛屾垨鑰呰〃绀烘墍鏈夎В鐨勪竴鑸舰寮忋
绛旓細鎵浠ユ牴鎹棤绌峰皬鍥涘垯杩愮畻鐨勬ц川銆傛棤绌峰皬閲忎箻浠ユ湁鐣屽彉閲=鏃犵┓灏忛噺銆傚嵆x瓒嬩簬0鏃,limxcos1/x=0銆傝В鍐抽棶棰樼殑鏋侀檺鎬濇兂 鏋侀檺鎬濇兂鏂规硶鏄暟瀛﹀垎鏋愪箖鑷冲叏閮ㄩ珮绛夋暟瀛﹀繀涓嶅彲灏戠殑涓绉嶉噸瑕佹柟娉曪紝涔熸槸鏁板鍒嗘瀽涓庡湪鍒濈瓑鏁板鐨勫熀纭涓婃湁鎵垮墠鍚悗杩炶疮鎬х殑銆佽繘涓姝ョ殑鎬濈淮鐨勫彂灞曘傛暟瀛﹀垎鏋愪箣鎵浠ヨ兘瑙ｅ喅璁稿鍒濈瓑鏁板鏃犳硶瑙ｅ喅鐨...
绛旓細cos(1/x)鏄湁鐣屽嚱鏁帮紱arctanx鏄湁鐣屽嚱鏁般傛湁鐣屽嚱鏁板畾涔夛細璁緁(x)鏄尯闂碋涓婄殑鍑芥暟锛岃嫢瀵逛簬浠绘剰鐨剎灞炰簬E锛屽瓨鍦ㄥ父鏁癿銆丮锛屼娇寰梞鈮(x)鈮锛屽垯绉癴(x)鏄尯闂碋涓婄殑鏈夌晫鍑芥暟銆傚叾涓璵绉颁负f(x)鍦ㄥ尯闂碋涓婄殑涓嬬晫锛孧绉颁负f(x)鍦ㄥ尯闂碋涓婄殑涓婄晫銆傚洜涓-1鈮os(1/x)鈮1锛屾寜鐓т笂闈㈠畾涔夛紝cos(1/x)鏄...
绛旓細閭ｅソ锛屽洖鍒板師棰橈紝鏃㈢劧cos(1/x)娌℃湁鏋侀檺鑰屼笖|cos(1/x)|<=1锛岄偅鎴戜滑鍙璁﹛^a->0锛岃繖鏍风敱浜庯細0 <= |x^a*cos(1/x)| = |x^a||cos(1/x)| <= |x^a|锛岃屼笖|x^a|瓒嬩簬0锛屾墍浠x^a*cos(1/x)|瓒嬩簬0锛屼篃灏辨槸x^a*cos(1/x)瓒嬩簬0 鎵浠ュ師棰樺彧瑕佽x^a->0灏卞彲浠ヤ簡...
绛旓細=位x^(位-1)*cos锛1/x)+x^(位-2)*sin(1/x)褰撐-2>0,x瓒嬩簬0鏃讹紝lim位x^(位-1)*cos锛1/x)+x^(位-2)*sin(1/x)=0 鑰屾寜瀹氫箟锛歭im(f(x)-f(0))/(x-0)=limx^位*cos锛1/x)/x=limx^(位-1)*cos锛1/x)=0锛埼-1>0)鎵浠ワ細褰撐-2>0鏃讹紝鍏跺鏁板湪x=0澶勮繛缁 ...
绛旓細cos(x+1)璺cosx閮芥槸鍩烘湰鍒濈瓑鍑芥暟銆俢os(x+1)璺焎osx鐨勫畾涔夊煙閮芥槸x鈭圧锛屽煎煙閮芥槸-1锛滐紳y锛滐紳1銆傝寰楀埌鍑芥暟y=cos锛坸+1锛夌殑鍥捐薄锛屽彧瑕佸皢鍑芥暟y=cosx鐨勫浘璞″悜宸﹀钩绉1涓崟浣嶅嵆鍙
绛旓細=3+limit(x*cos(1/x))=3 鍒╃敤娲涘繀杈炬硶鍒欐眰鏈畾寮忕殑鏋侀檺搴旀敞鎰忥細鈶犲湪鐫鎵嬫眰鏋侀檺浠ュ墠锛岄鍏堣妫鏌ユ槸鍚︽弧瓒0/0鎴栤垶/鈭炲瀷鏈畾寮忥紝鍚﹀垯婊ョ敤娲涘繀杈炬硶鍒欎細鍑洪敊銆傚綋涓嶅瓨鍦ㄦ椂锛堜笉鍖呮嫭鈭炴儏褰級锛屽氨涓嶈兘鐢ㄦ礇蹇呰揪娉曞垯锛岃繖鏃剁О娲涘繀杈炬硶鍒欎笉閫傜敤锛屽簲浠庡彟澶栭斿緞姹傛瀬闄愩傛瘮濡傚埄鐢ㄦ嘲鍕掑叕寮忔眰瑙ｃ傗憽鑻ユ潯浠剁鍚...
绛旓細楂樼瓑鏁板涔︼紙鍚屾祹鍏増涓婂唽43椤碉級涓婃湁杩欎箞涓涓畾鐞嗭細鏈夌晫鍑芥暟涓庢棤绌峰皬鐨勬垚缁╂槸鏃犵┓灏.褰揦瓒嬭繎浜0鏃,x鐨勫钩鏂硅嚜鐒惰秼杩戜簬0,鑰宑os锛1/x锛夋槸鏈夌晫鍑芥暟,鍏跺间粙浜庢璐1涔嬮棿,鏁呯敱瀹氱悊鐭鈫0鏃秞^2*cos(1/x)=0锛涙晠缁撴灉鏄痻鈫0鏃秞^2*cos(1/x)+1=1 ...
