f(x)在x=0处三阶可导求limx→0 {f '(x)/x2}=1时能用洛比达法则吗 设f(x)具有连续的二阶导数。点[0,f(0)]为曲线y=f...
\u8bbef(x)\u5728x=0\u5904\u8fde\u7eed\uff0c\u4e14lim(x\u8d8b\u4e8e0\uff09f(x)/x^2=1 ,\u8bc1\u660e\u51fd\u6570f(x)\u5728x=0\u5904\u53ef\u5bfc\u4e14\u53d6\u5f97\u6781\u5c0f\u503c\u3002f\uff08x\uff09\u5728x=0\u5904\u7684\u5bfc\u6570\u4e3af\u2018\uff080\uff09=lim(x\u8d8b\u4e8e0\uff09[f(x)-f(0)]/x
\u56e0\u4e3af(x)\u5728x=0\u8fde\u7eed\uff0c\u4e14lim(x\u8d8b\u4e8e0\uff09f(x)/x^2=1\uff0c\u6240\u4ee5f\uff080\uff09=0
lim(x\u8d8b\u4e8e0\uff09[f(x)-f(0)]/x=lim(x\u8d8b\u4e8e0\uff09f(x)/x
lim(x\u8d8b\u4e8e0\uff09f(x)/x^2=1,\u8bf4\u660ef\uff08x\uff09\u5728x=0\u5904\u4e8ex^2\u662f\u7b49\u4ef7\u65e0\u7a77\u5c0f
\u6240\u4ee5lim(x\u8d8b\u4e8e0\uff09f(x)/x=lim(x\u8d8b\u4e8e0\uff09x^2/x=x=0\uff0c\u8bc1\u660ef\uff08x\uff09\u5728x=0\u53ef\u5bfc\uff0c\u5207f \u2019 \uff08x\uff09=x
\u5f53x\u300a0\u65f6\uff0cf \u2019 \uff08x\uff090\u65f6\uff0cf \u2019 \uff08x\uff09>0\uff0c\u8bf4\u660ef(x)\u5728x=0\u53d6\u6781\u5c0f\u503c
\u4f60\u7684\u610f\u601d\u662f\u7528\u6d1b\u6bd4\u8fbe\u6cd5\u5219\uff0c\u8fd9\u662f\u53ef\u4ee5\u7684\u3002\u56e0\u4e3a\u4e8c\u9636\u5bfc\u6570\u5b58\u5728\u3002
绛旓細鐞嗙敱锛歭im f'(x)锛漧im f'(x)/x^2*x^2锛漧im f'(x)/x^2 *lim x^2=1*0=0.
绛旓細[f锛0+3h锛-f锛0锛+f锛0锛-f锛0-h锛塢/2h =[f锛0+3h锛-f锛0锛塢/2h+[锛坒锛0锛-f锛0-h锛塢/2h 鐢卞鏁板畾涔夊彲鐭im(h瓒嬩簬0)[f锛0+3h锛-f锛0锛塢/2h=锛3/2锛塮'锛0锛塴im(h瓒嬩簬0)[锛坒锛0锛-f锛0-h锛塢/2h=锛1/2锛塮'锛0锛変簬鏄師寮=2f'锛0锛...
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绛旓細f(x)鏈3闃杩炵画瀵兼暟锛屽熀鏈笂浣滅敤鍜屾剰涔夊緱缁撳悎鍏蜂綋闂 鐒跺悗鍙堥拡瀵规х殑鍒嗘瀽鍜屽鐞嗭紝浣嗗叾涓渶閲嶈鐨勬槸 瀵兼暟鐨勫畾涔夊拰杩炵画鐨勬蹇靛紕娓呮灏变笉浼氭湁澶ぇ鐨勯棶棰 鐒跺悗f(x)鍦▁=0鐨勯偦鍩熷唴鍙锛屽嵆f'''(x)锛(x鈭圖)瀛樺湪锛孌涓簒=0鐨勬煇棰嗗煙 閭ｄ箞蹇呯劧鏈変綆闃剁殑瀵煎嚱鏁版槸杩炵画鐨勶紝鍗砯'(x)锛宖''(x)鍦▁鈭圖...
绛旓細涓嶄竴瀹氱殑锛屾瘮濡傝x鐨5/2娆℃柟婊¤冻鏉′欢锛屼絾涓夐樁瀵兼暟鍦0涓嶈繛缁紝鍥犱负鏃犲畾涔
绛旓細鍙兘鐢ㄩ害鍏嬪姵鏋楀叕寮忥紝瑙ｉ濡備笅锛氬F(x)鍦▁=0澶鐢ㄩ害鍏嬪姵鏋楀叕寮忓睍寮锛欶(x)=F(0)+F'(0)*x+F''(0)*x^2/2+F'''(A)*x^3/3!锛孉鈭(0,1)鈥︹(1)鍙堟湁F(x)=x^3*f(x)鏁匜(0)=F(1)=0 浠ｅ叆(1)寮忥細F(x)=0+0+0+F'''(A)*x^3/3!鍐嶄护x=1锛屾湁锛欶(1)=F'''(A)/6...
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绛旓細鍥炵瓟锛歺瓒嬩簬0鐨勬椂鍊,1-cosx鍜寈^2涔熻秼浜0, 閭ｄ箞鍒嗘瘝tan(x^2)灏辩瓑浠蜂簬x^2, 鎵浠 鍘熸瀬闄 =lim(x->0) f(1-cosx) / x^2 浣跨敤娲涘繀杈炬硶鍒,瀵瑰垎瀛愬垎姣嶅悓鏃舵眰瀵 =lim(x->0) f '(1-cosx) *(1-cosx)' / 2x 鏄剧劧(1-cosx)'= sinx 閭ｄ箞 鍘熸瀬闄 =lim(x->0) f '(1-cosx
绛旓細[f(x鏂)]'=2xf'(x^2)[f(x鏂)]''=2f'(x^2)+4x^2f''(x^2)[f(x鏂)]'''=4xf''(x^2)+8xf''(x^2)+8x^3f'''(x^2)
