六屆團(tuán)支書(shū)聯(lián)席會(huì)復(fù)習(xí)寶典級(jí)上學(xué)期高數(shù)老師版

上節(jié)內(nèi)容回δ0,0|x上節(jié)內(nèi)容回δ0,0|xx0|δ時(shí),|f(xA|limf(x)=A:?ε>(1)f(x)0|xx0|δ時(shí)是有界的A(2)若A>0,取ε ,則在0<|x?x|<δ02Af(xAε 保2在U(0,δ)內(nèi)成立.2/函數(shù)極限的運(yùn)算法變量代冪指函數(shù)的極作業(yè)3/函數(shù)極限的運(yùn)算法變量代冪指函數(shù)的極作業(yè)3/函數(shù)極限函數(shù)極限的性關(guān)于函數(shù)極限的三個(gè)問(wèn)題唯一性局部有界性局部保號(hào)性存在性、唯一性、計(jì)算方法4/函數(shù)極函數(shù)極限的性關(guān)于函數(shù)極限的三個(gè)問(wèn)題:存在性、唯一性、計(jì)算方法唯一性局部有界性局部保號(hào)性4/四則運(yùn)定理limf(x)limg(四則運(yùn)定理limf(x)limg(x均存在.()limf(x)+g(x)=limf(x)+limlimf(x)·g(x)=limf(x)·lim() lim=(limg(x)?= lim推 limc1f(x)+c2g(x)=c1limf(x)+c2lim稱為函數(shù)極限的線性性質(zhì)5/四則運(yùn)定理limf(x)limg(x均存在.lim(f四則運(yùn)定理limf(x)limg(x均存在.lim(f(x)+g(x))=limf(x)+lim()lim)·(x)=limf(x)·lim lim(limg(x)?== lim推()limcf(x)+c)=climf(x)+clim1212稱為函數(shù)極限的線性性質(zhì)5/問(wèn)4.2(多項(xiàng)式的極限P(xa0xna1xn?1···an,a0?0,n問(wèn)4.2(多項(xiàng)式的極限P(xa0xna1xn?1···an,a0?0,n求limP(x)與lim 題型: 例(1)∞?∞型極不能直接用極限的四則運(yùn).(5x+0,型. (2x?3)2(3x+(2).為0或∞的因子,再用極限的四則運(yùn)2.型.提取無(wú)窮因子,化∞·A型.若A?=0,則原極限為5(5x+xn?(3).xm—1() (4)?.若A=0,則再化 型0x? x2?06/4.2(多項(xiàng)式的極限P(xa0xna1xn?1···an,a0?0,n求limP(x)與4.2(多項(xiàng)式的極限P(xa0xna1xn?1···an,a0?0,n求limP(x)與lim例(1) 題型:∞型極 (2x?3)2(3x+.不能直接用極限的四則運(yùn)算(5x+型.先消去分子、分母中極0,(2x?3)2(3x+ (2).為0或∞的因子,再用極限的四(5x+5算法則xn?(3).2.∞型.提取無(wú)窮因子,化x→1m—()∞·A型.若A?=0,則原極限為 (4).?A0,則再化型x?x2?006/變量代例4.4 )3x?變量代例4.4 )3x?3x?— +1問(wèn)4.5(復(fù)合函數(shù)的極限或變量代換limf(xA,limφ(tx0.是否一定limf[φ(tA答.若滿足以下條件之一,則結(jié)論成立(1)δ0>0,當(dāng)0<|tt0|<δ0時(shí),φ(t?(2)limf(x)=7/變量代例4.4 ()33—變量代例4.4 ()33— +1x?x?4.5(復(fù)合函數(shù)的極限變量代換設(shè)limf(xA,limφ(tx0.是否一定有l(wèi)imf[φ(tA答.若滿足以下條件之一,則結(jié)論成立(1)δ0>0,當(dāng)0<|tt0|<δ0時(shí),φ(t)?=(2)limf(x)=7/例 n1+x?(1).x例 n1+x?(1).x(1+1n1—1?(2).8/例(1)n1+x?.x1(2)例(1)n1+x?.x1(2)lim(1+x)—1?1.n8/冪指函數(shù)的極在極限的計(jì)算中 經(jīng)常出現(xiàn)下面的問(wèn)題lim冪指函數(shù)的極在極限的計(jì)算中 經(jīng)常出現(xiàn)下面的問(wèn)題limf(xA,limg(xB.是否一定limf(x)g(xAB答.A0,AB均有限,則結(jié)論成立(1)A=B=(2)A=+∞,B=例.任取a1,(1型=ax0∞1)型(3)A=1,B==.1型lim(ax)=9/冪指函數(shù)的極在極限的計(jì)算中,經(jīng)常出現(xiàn)下面的問(wèn)題問(wèn)limf冪指函數(shù)的極在極限的計(jì)算中,經(jīng)常出現(xiàn)下面的問(wèn)題問(wèn)limf(xA,limg(xB.是否一定有l(wèi)imf(x)g(xAB答.A0,AB均有限,則結(jié)論成立以下三種情形是“未定式(1)A=B=例.任取a1,(x1型 =,x(2)A=+∞,B=1