高等數(shù)學(xué)常用極限求法[1]1

1、1一、求函數(shù)極限的方法一、求函數(shù)極限的方法1、運(yùn)用極限的定義例:用極限定義證明:1223lim22?????xxxx證:由244122322????????xxxxxx??2222?????xxx取則當(dāng)時就有0??????????20x??????12232xxx由函數(shù)極限定義有:???1223lim22?????xxxx2、利用極限的四則運(yùn)算性質(zhì)若Axfxx??)(lim0Bxgxx??)(lim0(I)?????)()(lim0xg

2、xfxx)(lim0xfxx??BAxgxx???)(lim0(II)??BAxgxfxgxfxxxxxx????????)(lim)(lim)()(lim000(III)若B≠0則：BAxgxfxgxfxxxxxx?????)(lim)(lim)()(lim000（IV）（c為常數(shù)）cAxfcxfcxxxx??????)(lim)(lim0035、利用無窮小量性質(zhì)法（特別是利用無窮小量與有界量之乘積仍為無窮小量的性質(zhì)）設(shè)函數(shù)f(x)、

3、g(x)滿足：（I）0)(lim0??xfxx(II)(M為正整數(shù))Mxg?)(則：0)()(lim0??xfxgxx例:求xxx1sinlim0??解:由而0lim0??xx11sin?x故原式=01sinlim0???xxx6、利用無窮小量與無窮大量的關(guān)系。（I）若：則??)(limxf0)(1lim?xf(II)若:且f(x)≠0則0)(lim?xf??)(1limxf例:求下列極限①②51lim???xx11lim1??xx解: