limx趋近于01-cosx/x=多少 limx→0cosx/x=

limx\u8d8b\u5411\u4e8e0 \uff081-cosx\uff09/x²

limx\u8d8b\u5411\u4e8e0 \uff081-cosx\uff09/x²\u7684\u7ed3\u679c\u7b49\u4e8e1/2\u3002
\u89e3\uff1alim(x\u21920)(1-cosx)/(x^2) \uff08\u6d1b\u5fc5\u8fbe\u6cd5\u5219\uff0c\u5206\u5b50\u5206\u6bcd\u540c\u65f6\u6c42\u5bfc\uff09
=lim(x\u21920)sinx/(2x) \uff08\u6d1b\u5fc5\u8fbe\u6cd5\u5219\uff0c\u5206\u5b50\u5206\u6bcd\u540c\u65f6\u6c42\u5bfc\uff09
=lim(x\u21920)cosx/2
=1/2
\u5373lim(x\u21920)(1-cosx)/(x^2)\u7684\u6781\u9650\u503c\u7b49\u4e8e1/2\u3002
\u6269\u5c55\u8d44\u6599\uff1a
1\u3001\u6781\u9650\u8fd0\u7b97\u6cd5\u5219
\u4ee4limf(x)\uff0climg(x)\u5b58\u5728\uff0c\u4e14\u4ee4limf(x)=A\uff0climg(x)=B\uff0c\u90a3\u4e48
\uff081\uff09\u52a0\u51cf\u8fd0\u7b97\u6cd5\u5219
lim(f(x)\u00b1g(x))=A\u00b1B
\uff082\uff09\u4e58\u6570\u8fd0\u7b97\u6cd5\u5219
lim(a*f(x))=a*limf(x)\uff0c\u5176\u4e2da\u4e3a\u5df2\u77e5\u7684\u5e38\u6570\u3002
2\u3001\u6c42\u6781\u9650\u7684\u65b9\u6cd5
\uff081\uff09\u5206\u5b50\u5206\u6bcd\u6709\u7406\u5316
\uff082\uff09\u5939\u903c\u6cd5\u5219
3\u3001\u6781\u9650\u7684\u91cd\u8981\u516c\u5f0f
\uff081\uff09lim(x\u21920)sinx/x=1\uff0c\u56e0\u6b64\u5f53x\u8d8b\u4e8e0\u65f6\uff0csinx\u7b49\u4ef7\u4e8ex\u3002
\uff082\uff09lim(x\u21920)(1+x)^(1/x)=e\uff0c\u6216\u8005lim(x\u2192\u221e)(1+1/x)^x=e\u3002
\uff083\uff09lim(x\u21920)(e^x-1)/x=1\uff0c\u56e0\u6b64\u5f53x\u8d8b\u4e8e0\u65f6\uff0ce^x-1\u7b49\u4ef7\u4e8ex\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u6781\u9650

\u5206\u6bcd\u6781\u9650\u4e3a0\uff0c
\u5012\u8fc7\u6765\uff0c

lim(x\u21920)x/cosx
=0/cos0
=0/1
=0
\u2234x\u21920\u65f6\uff0c
x/cosx\u662f\u65e0\u7a77\u5c0f\uff0c
\u2234cosx/x\u662f\u65e0\u7a77\u5927\uff0c

lim(x\u21920)cosx/x=\u221e

极限是0，如下

limx瓒嬭繎浜0(1-cosx)/ln(1-x²)涓轰粈涔堜笉鑳界敤娲涘繀杈?
绛旓細鍙互鐢ㄣ備絾鏄病蹇呰鑷壘楹荤儲銆傘傚垎姣嶇瓑浠蜂簬璐x骞虫柟锛屽垎瀛愮瓑浠蜂簬浜屽垎涔媥²銆傜害鍒嗗緱鍒-1/2=-0.5銆傘備妇鎶ユ暟瀛楀笣鍥紾G娉涙互浣嗘槸鏄竴涓绠楀櫒缃戦〉銆傘傚緢涓嶉敊浜嗐傘傘

姹limx瓒嬪悜浜0,x鍒嗕箣1-cosx,姹傝繃绋
绛旓細鍥犱负x瓒嬪悜浜0锛屽彲鐭ュ畠鏄0/0鍨嬶紝鐢ㄦ礇蹇呰揪娉曞垯锛屼笂涓嬪悇姹傚锛屽緱鍑limx瓒嬪悜浜0锛1/sinx锛夛紝灏0浠ｈ繘鍘伙紝寰楀嚭1/鏃犵┓灏=鏃犵┓澶р垶

涓轰粈涔x瓒嬩簬闆鏃cosx-1鍜-(x;
绛旓細cosx-1鍜-(x²)/2鏄瓑浠锋棤绌峰皬閲忋傝В锛歝osx鍦x0=0澶勫睍寮寰梒osx=1-x²/2+x⁴/4-x⁶/6+...+(-1)ⁿx²ⁿ/2n... 锛屽嵆1-cosx=x²/2-x⁴/4+x⁶/6+...+(-1)ⁿx²ⁿ/2n...锛屾墍浠lim[(1-cosx)...

limx瓒嬪悜浜0,(1-cosx)/xsinx
绛旓細lim(x鈫0)(1-cosx)/xsinx =lim(x鈫0)(x²/2)/x²=1/2

褰x瓒嬭繎浜0鏃,cosx/1-x鐨勬瀬闄愮瓑浜涓銆俢osx涓1-x涓虹瓑浠锋棤绌峰皬鍚?
绛旓細鏄x瓒嬩簬0鏃剁殑绛変环鏃犵┓灏忋

姹備笅鍒楁瀬闄 lim x->0 (1-cosx)/(sinx)
绛旓細鐐瑰嚮鏀惧ぇ锛屽啀鐐瑰嚮鍐嶆斁澶с

limx瓒嬭繎浜0 1-cosx闄や互x涔樹互cosx
绛旓細濡傚浘

lim1/1-cosx褰x瓒嬭繎浜0鏃舵眰鏋侀檺骞惰鏄庣悊鐢
绛旓細鍥犱负x瓒嬭繎浜0鏃禼osx鏋侀檺涓1锛1-cosx鏋侀檺涓0锛屾墍浠1/1-cosx褰搙瓒嬭繎浜0鏃舵瀬闄愪负鏃犵┓澶

lim x鈫0 鈭1-cosx²/1-cosx
绛旓細x瓒嬩簬0鐨勬椂鍊欙紝1-cosx绛変环浜0.5x²浜庢槸杩欓噷鐨1-cosx²绛変环浜0.5(x²)²鍐嶅紑鏍瑰彿鍗斥垰1-cosx²绛変环浜1/鈭2 *x²鎵浠ュ緱鍒板師鏋侀檺=lim(x瓒嬩簬0) (1/鈭2 *x²) / 0.5x²=2/鈭2= 鈭2 ...

x瓒嬪悜浜0,1-cos2x鏄灏?
绛旓細x瓒嬩簬0鏃讹紝1-cos2x绛変环浜2x^2.