设函数f（x）在点x0处可导，试求下列各极限的值．（1）lim△x→0f(x0?△x)f(x0)△x；（2）limh→0f(x0+h) 设函数f(x)在点Xo处可导，求下列极限lim(ΔX→0)[...
\u8bbe\u51fd\u6570f\uff08x\uff09\u5728\u70b9x 0 \u5904\u53ef\u5bfc\uff0c\u8bd5\u6c42\u4e0b\u5217\u5404\u6781\u9650\u7684\u503c\u3002\uff081\uff09 \uff1b\uff082\uff09 \u3002\u89e3\uff1a\uff081\uff09\u539f\u5f0f= =-f\u2032\uff08x 0 \uff09\uff08\u25b3x\u21920\u65f6\uff0c-\u25b3x\u21920\uff09 \uff082\uff09\u539f\u5f0f= \u3002
lim(\u0394X\u21920)[F(X)-F(Xo-\u0394X)]/\u0394X
=lim(\u0394X\u21920)[F(X)-F(Xo-\u0394X)]/\u3010-\u0394X*\uff08-1\uff09\u3011
=-f'(x0)
| lim |
| △x→0 |
| f(x0?△x)?f(x0) |
| ?(?△x) |
=-
| lim |
| △x→0 |
| f(x0?△x)?f(x0) |
| ?△x |
（2）
| lim |
| h→0 |
| f(x0+h)?f(x0?h) |
| 2h |
=
| 1 |
| 2 |
| lim |
| h→0 |
| f(x0+h)?f(x0)+f(x0)?f(x0?h) |
| h |
=
| 1 |
| 2 |
| lim |
| h→0 |
| f(x0+h)?f(x0) |
| h |
| f(x0?h)?f(x0) |
| ?h |
=
| 1 |
| 2 |
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绛旓細鍘熷紡=lim[xf(x0)-xf(x)+xf(x)-x0f(x)]/(x-x0)=lim[xf(x0)-xf(x)]/(x-x0)+lim[xf(x)-x0f(x)]/(x-x0)=limx[f(x0)-f(x)]/(x-x0)+f(x0)=-x0f'(x0)+f(x0)=f(x0)-x0f'(x0)
绛旓細杩欐槸姝ｇ‘鐨勩傚鏋滃畠鍦ㄧ偣X0澶勮繛缁紝鍒鍑芥暟f(x)鍦ㄧ偣x0澶蹇呭畾鍙銆傞敊璇,姣斿f(x)=x鐨勭粷瀵瑰硷紝鍦▁o=0鏃朵笉杩炵画锛屽洜涓哄畠鐨勫乏鍙虫瀬闄愪笉鐩哥瓑銆
绛旓細lim(x--xo)=|f(x)|-|f(x0)|/x=lim(x-->x0)|f(x)|/x 鎵浠im(x-->x0+)|f(x)|/x =f`(x0)lim(x-->x0-)|f(x)|/x =-f`(x0)鍥犱负f`(x0)涓嶇瓑浜0,鍗冲乏鍙瀵兼暟涓嶇浉绛夛紝鎵浠ヤ笉鍙
绛旓細lim(鈻硏->0)(f(x0-2鈻x)-f(X0))/鈻硏)=lim(鈻硏->0) -1/2*(f(x0-2鈻硏)-f(X0))/(-2鈻硏)=-1/2f'(x0)=-a/2
绛旓細3F'(x0),鍙皢鍒嗘瘝涔3锛屽彉鎴F(x)鍦▁0瀵兼暟鐨勫畾涔夊紡銆傜粨鏋滄槸璇鐐瑰鏁鐨3鍊
绛旓細寤朵几瑙ｉ噴锛氭暟瀛﹂棶棰橀鍏堜粠瀹氫箟鍏ユ墜锛岄鍏堣繛缁殑姒傚康鏄嚱鏁帮細 鍑芥暟f(x)鍦ㄧ偣 鐨勬煇涓偦鍩熷唴鏈夊畾涔夛紝濡傛灉鏈 锛屽垯绉板嚱鏁板湪鐐 澶勮繛缁紝涓旂О 涓哄嚱鏁扮殑鐨勮繛缁偣銆傝瀵兼暟鐨勫畾涔夋槸锛璁惧嚱鏁y=f(x)鍦ㄧ偣x0鐨勬煇涓偦鍩熷唴鏈夊畾涔夛紝褰撹嚜鍙橀噺x鍦▁0澶鏈夊閲徫x锛(x0+螖x)涔熷湪璇ラ偦鍩熷唴鏃讹紝鐩稿簲鍦板嚱鏁...
绛旓細(x0+h)-(x0-h)=2h 鍥犳鏍规嵁鏋侀檺鐨勫畾涔夊緱 limf(x0+h)-f(x0-h)/2h=f'(x0)
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绛旓細鏄剧劧鏄敊鐨勶紝璇︽儏濡傚浘鎵绀
