第1講函數(shù)的極限、連續(xù)性2015基礎(chǔ)預(yù)習(xí)

2015工 會(huì)判斷間斷點(diǎn)的類型 在自變量的變化過(guò)程中,如果對(duì)應(yīng)的函數(shù)值無(wú)限接近于f(x)函數(shù)的極限有兩種變化形式xx或稱為兩種趨近過(guò)程x指:x及xxx指:xx及x 設(shè)函數(shù)f(x)當(dāng)x大于某一正數(shù)時(shí)有定義.如果存在著正數(shù)X,使得當(dāng)x滿足不等式x X時(shí),對(duì)應(yīng)的f(x)f(xAεf(x)當(dāng)x記作limf(x)limf(x)Alimf(x)limf(x) limf(x)AyAε(ε0X0,xX時(shí)yf(x)的圖像夾在yAε兩直線之間，如圖yyA+ε?oXxxx0f(x設(shè)函數(shù)yf(x)在x0的某個(gè)去心鄰域內(nèi)有定義若ε0(不論多么小),δ0,當(dāng)0xx0δ時(shí)f(xAAf(x)當(dāng)xx0時(shí)的極限記作:limf(x)A此時(shí)也稱當(dāng)xx0時(shí),f(x)的極限存在否則,稱當(dāng)xx0時(shí),f(x)的極限不存在limf(x)AxyAε(ε0存在δ0,x^(x0δ內(nèi)時(shí),yf(x)^yyy=fAox設(shè)f(x)在x0的左邊附近(右邊附近)有定義使得當(dāng)0x0xδ(0xx0δ)時(shí)f(x)AA為,當(dāng)xx0時(shí)的左極限(右極限0記作limf(x)Alimf(x)00xx0

xx

limf(x)alimf(x)limf(x)

f(x)在x0點(diǎn)極限存在的充要條件是 x例:f(x)cos

x

討論limf(x)若limf(x)0,則稱f(x)例:lim10limsinx0limcosxx 21sinxcosx是相應(yīng)過(guò)程的無(wú)窮小量注意無(wú)窮小量與極限過(guò)程不能分開(kāi)不能脫離過(guò)程談例如limsinx0是無(wú)窮小量;limsinx1 2若α(x)是某個(gè)極限過(guò)程的無(wú)窮小量,f(x)是該過(guò)程的有界量,則α(xf(x)是該過(guò)程例：求lim1sinx設(shè)某過(guò)程,limf(x)0limg(x)0且limf(x)g(x)l0f(x)g(x)的高階無(wú)窮小量l0f(x)與g(x)l1f(x)與g(x)為等價(jià)無(wú)窮小量limf(x)l0f(x)是g(x)的k階無(wú)窮小;g(x)kx

sinx~xtanx~ arcsinx~ arctanx~ex1~ ln1x~21cosx~1x221 1 1~1xsinx x316若limf(x)Alimg(x)B,limf(x)g(x)limf(x)limg(x)Alimf(x)g(x)limf(x)limg(x)A

若B0limf(x)limf(x)g(x) limg(x) 若limf(x)A,則在該過(guò)f(x)Aα(x)α(x)是該過(guò)程的無(wú)窮小量有限個(gè)無(wú)窮小的積極也是無(wú)窮小求

11x

x2

x2x2求

4x32x23x2

limsinx

1 1x

xx

(lim1 11求lim(cosxx2x求limx x0型:limfx0limg(x)0型:limf(x)limg(x)fxgx都存在,且limfxg則

fgx求limxsin 設(shè)函數(shù)f(x)在x0某鄰域U(x0內(nèi)有定義limf(xf(x0則稱函f(x)在x0點(diǎn)連續(xù)x0f(x的連續(xù)點(diǎn);否則f(x)在x0間斷,x0f(x)的間斷點(diǎn)或不連續(xù)點(diǎn);f(x)在x右鄰域U(x或左領(lǐng)域U(x－)) 內(nèi)有定義limf(x)f(x0limf(x)f(x0 xx f(x)在x0點(diǎn)連續(xù) f(x)在x0點(diǎn)左連續(xù)且右連f(x|x

f(x)

xx

當(dāng)x0f(x)在x0點(diǎn)是否連續(xù)定理.(1初等函數(shù)在其定義域內(nèi)連續(xù)

x2ln(4 x2 x0 cosxlimf(x)f(x0)要使它成立f(x)x0有定義f(x)x0的極限存在這三條有一條不成立f(x)x0不連續(xù)(間斷fx)

當(dāng)x0時(shí)當(dāng)x0時(shí),的間斷點(diǎn).當(dāng)x0時(shí),可去間斷點(diǎn):跳躍間斷點(diǎn):斷點(diǎn)。左,右極限中至少有一個(gè)不存在的間斷點(diǎn)無(wú)窮間斷點(diǎn):求ysin1的間斷點(diǎn), x根的存在定理f