(cosx-1)/x的极限？
\uff081\uff0dcosx\uff09\u00f7x\u2227p\u7684\u6781\u9650\u503c\u4e0ep\u6709\u5173\uff0c\u800c\u4e14\u4f60\u6ca1\u8bf4x\u662f\u8d8b\u8fd1\u4e8e\u54ea\u4e2a\u6781\u9650\uff01\u662f0\uff1f\u8fd8\u662f\u65e0\u7a77\uff1f
\u662f\u96f6\u7684\u8bdd\uff0c\u5f53p\u5927\u4e8e2\uff0c\u6781\u9650\u662f\u65e0\u7a77\u5927\uff0c\u5f53p\u5c0f\u4e8e2\uff0c\u6781\u9650\u662fo\uff0c\u7b49\u4e8e2\u662f1\uff0f2
此极限为“0”。
洛必达法则或者泰勒展开。
绛旓細cosx-1=-2sin(x/2)^2 濡備笅锛歝osx-1=cos(x/2+x/2)-1 =[cos(x/2)]^2-[sin(x/2)]^2-([cos(x/2)]^2+[sin(x/2)]^2)=-2sin(x/2)^2 鍚岃涓夎鍑芥暟鐨勫熀鏈叧绯诲紡 鍊掓暟鍏崇郴锛歵an伪 路cot伪=1銆乻in伪 路csc伪=1銆乧os伪 路sec伪=1銆傚晢鐨勫叧绯伙細 sin伪/cos伪=tan伪=sec...
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绛旓細1-cosx = 2sin²(x/2)鐢ㄤ簩鍊嶈鍏紡锛歝os2a=1-2sin²a 1-cos2a=2sin²a 鎵浠ワ細1-cosx=2sin²(x/2)~2脳(x/2)²~x²/2 鎵浠ワ細1-cosx鐨勭瓑浠锋棤绌峰皬涓簒²/2 浜屽嶈鍏紡閫氳繃瑙捨辩殑涓夎鍑芥暟鍊肩殑涓浜涘彉鎹㈠叧绯绘潵琛ㄧず鍏朵簩鍊嶈2伪鐨勪笁瑙掑嚱鏁板硷紝浜...
绛旓細cosx鍑涓鐨勭瓑浠锋棤绌峰皬鏄痻²/2銆傜敤浜屽嶈鍏紡锛歝os2a=1-2sin²a锛1-cos2a=2sin²a锛屾墍浠1-cosx=2sin²锛坸/2锛墌2脳锛坸/2锛²~x²/2锛屾墍浠1-cosx鐨勭瓑浠锋棤绌峰皬涓簒²/2銆傜瓑浠锋棤绌峰皬鏄棤绌峰皬涔嬮棿鐨勪竴绉嶅叧绯伙紝鎸囩殑鏄湪鍚屼竴鑷彉閲忕殑瓒嬪悜杩囩▼涓紝鑻ヤ袱涓...
绛旓細cosx/2)^2-1鎴朿osx=1-(sinx/2)^2, 杩欓噷鏄剧劧瑕佺敤鍚庤咃紝cosx-1=-(sinx/2)^2, 濡傛灉缁撴灉鏄-4鍒嗕箣1鐨勮瘽锛岃鏄巗inx/2=姝ｈ礋1/2锛屽鏋滅瓑浜1/2, 閭ｄ箞x灏辨槸pi/3锛屽鏋滄槸璐1/2锛岄偅涔坸灏辨槸-pi/3銆傚掓帹鐨勭粨鏋滆繕鍙互鍔犱笂2kpi, 鍥犱负sin鏄懆鏈熸х殑锛屼絾涓鑸嚭棰樼敤鐨勯兘鏄繖涓や釜鍊笺
