等价无穷小代换的一道题 一道数学题，等价无穷小替换
\u6c42\u52a9\u5173\u4e8e\u7b49\u4ef7\u65e0\u7a77\u5c0f\u4ee3\u6362\u7684\u4e00\u9053\u9898\u76ee\u7b54\u6848\u662f\u4e0d\u5bf9\u7684\uff0c\u4f60\u8bf4\u5f97\u5bf9
\u5373\u539f\u5f0f=sin(1/sin(1-lim x->0 x^2/sin(1-cosx)))
\u5e94\u8be5\u76f4\u63a5\u5148\u8ba1\u7b97\u6781\u9650
lim x->0 x^2/sin(1-cosx)
\u56e0\u4e3a\u5269\u4e0b\u7684\u90e8\u5206\u90fd\u6ca1\u6709x
\u7b49\u4ef7\u91cf\u4ee3\u6362
=lim x->0 x^2/(1-cosx)
=lim x->0 x^2/(x^2/2)
=2
\u7136\u540e\u4ee3\u5165\u539f\u5f0f
=sin(1/sin(1-2))
=sin(1/sin(-1))
sinx =x-x^3/6
sinx -x ~ -x^3/6
(sinx-x)/x\u786e\u5b9e\u662f\u9ad8\u9636\u65e0\u7a77\u5c0f\uff0c\u4f46\u662f\u9898\u76ee\u89e3\u6cd5\u786e\u5b9e\u4e0d\u4e25\u8c28\uff0c\u4ed6\u6ca1\u6709\u63a8\u5bfc\u5230\u4e0a\u9762\u7ed3\u679c\u76f4\u63a5\u66ff\u4ee3\u662f\u4e0d\u5bf9\u7684
如图
如图
绛旓細姝ｇ‘鍋氭硶鏄厛閫氬垎锛屽啀鐢ㄦ礇蹇呰揪娉曞垯鎴朤aylor灞曞紡銆傜浜岄绫讳技锛(1+1/x)^x鐩稿鏁翠釜琛ㄨ揪寮忎笉鏄洜瀛愶紝鍥犳涓嶈兘绛変环鏇挎崲銆傛纭仛娉曟槸锛氬厛鍙栧鏁帮紝鐒跺悗鐢ㄦ礇蹇呰揪娉曞垯鎴朤aylor灞曞紡锛屽缓璁敤 Taylor灞曞紡銆傚彇瀵规暟鍚庯紝lim x*(1锛峹ln(1+1/x))锛漧im x*(1锛峹銆1/x锛1/(2x^2)+灏弌(1/x^2)銆)锛...
绛旓細瀵逛簬a-b杩欑寮忓瓙锛岃嫢a涓巄閮芥槸鏃犵┓灏忛噺涓攁涓巄涓嶇瓑浠凤紝鍒檃涓巄鍙互鍒嗗埆鐢ㄧ瓑浠锋棤绌峰皬杩涜浠ｆ崲銆傝繖閬撻鐩涓嚭鐜颁簡(e^(ax)-1)-ax杩欎釜寮忓瓙锛岀敱浜(e^(ax)-1)~ax锛屾墍浠ヤ笉鑳芥妸(e^(ax)-1)浠ｆ崲鎴恆x銆備竴鍙ヨ瘽鎬荤粨锛氬a卤b涓殑a鍜宐鍒嗗埆杩涜绛変环鏃犵┓灏忎唬鎹鏃讹紝蹇呴』淇濊瘉浠ｆ崲鍚巃卤b鐨勭粨鏋滀笉绛変簬0 ...
绛旓細褰撲负涔樼Н鏃跺彲鐢绛変环鏃犵┓灏忎唬鎹姹傛瀬闄 浣嗘槸褰撳姞鍑忔椂灏遍渶瑕佸厛璁＄畻 涓句釜渚嬪瓙 锛坰inx-tanx锛/x^3 x瓒嬭繎浜0鐨勬瀬闄 sinx=x+o1锛坸锛 tanx=o2锛坸锛塻inx-tanx=o1锛坸锛-o2锛坸锛=o锛坸锛塠o1锛坸锛塷2锛坸锛塷锛坸锛夐兘鏄痻楂橀樁鏃犵┓灏廬鍥犱负浜岃呯浉鍑忔妸宸茬煡鐨勯儴鍒嗛兘鎶垫秷鎺変簡 鍓╀笅鐨勯儴鍒嗘槸o锛坸锛夋槸涓涓...
绛旓細姝ら鏄珮绛夋暟瀛︿腑鍏充簬绛変环鏃犵┓灏忕殑棰樼洰锛屽叧閿偣灏辨槸绛変环鏃犵┓灏忕殑浠ｆ崲銆傜瓑浠锋棤绌峰皬閲屾湁杩欐牱涓涓叕寮忥細褰搙->0鏃(1+x)^a-1绛変环浜巃x 鎵浠ュ湪杩欓噷濂楃敤姝ゅ叕寮忋2娆℃牴鍙峰氨鐩稿綋浜0.5娆℃柟鍢涖(1+ax^2)^0.5-1绛変环浜1/2ax^2锛涘彟澶栦竴鏂归潰(sinx)^2绛変环浜巟^2杩欎釜鎴戜笉鐢ㄥ瑙ｉ噴浜嗗惂銆傚凡鐭ヤ袱鑰呬负绛変环...
绛旓細鍏堝寲绠锛屽啀鎹㈠厓锛屾渶鍚庡埄鐢绛変环鏃犵┓灏忔浛鎹鍗冲彲姹傚嚭缁撴灉銆
绛旓細浠ヤ笅lim绗﹀彿鍧囩渷鐣 鏍规嵁娲涘繀濉旀硶鍒,(1+cos蟺x)/(x-1)^2 =-蟺sin蟺x/(2x-2)=-蟺^2cos蟺x/2 =蟺^2/2
绛旓細灏辨槸sin((ax^2)/3)/(2x^2)=(a/6)(sin((ax^2)/3)/((ax^2)/3)->1 a=6
绛旓細绛旀鏄笉瀵圭殑锛屼綘璇村緱瀵 鍗冲師寮=sin(1/sin(1-lim x->0 x^2/sin(1-cosx)))搴旇鐩存帴鍏堣绠楁瀬闄 lim x->0 x^2/sin(1-cosx)鍥犱负鍓╀笅鐨勯儴鍒嗛兘娌℃湁x 绛変环閲浠ｆ崲 =lim x->0 x^2/(1-cosx)=lim x->0 x^2/(x^2/2)=2 鐒跺悗浠ｅ叆鍘熷紡 =sin(1/sin(1-2))=sin(1/sin(-1))
绛旓細(4)x->0 tanx = x+(1/3)x^3+o(x^3)sinx = x-(1/6)x^3+o(x^3)tanx -sinx =(1/2)x^3+o(x^3)鈭(1+tanx) -鈭(1+sinx)=(1+tanx) -(1+sinx) /[鈭(1+tanx) +鈭(1+sinx) ]=(tanx -sinx)/[鈭(1+tanx) +鈭(1+sinx) ]=[(1/2)x^3+o(x^3)]/2 =(1...
绛旓細鏋侀檺=-2ln2
