第四节极限的运算 米 ✉囧
第四节 极限的运算
定理 设m/八四=A8(=B, 则(0)Iim[f,±g(x]=limf(x)±lim8() x→X0 =A士B, ②)fW8=mf-s)=AB, 3)血Cf(x)=C.limf(x)=CA(C为常数 X→X0 ✉合
0 0 lim ( ) , lim ( ) , x x x x f x A g x B → → 设 = = 0 0 0 (1) lim ( ) ( ) lim ( ) lim ( ) ; x x x x x x f x g x f x g x A B → → → = = 0 0 0 (2) lim ( ) ( ) lim ( ) lim ( ) ; x x x x x x f x g x f x g x A B → → → = = 则 定理 0 0 lim ( ) lim ( ) ( x x x x Cf x C f x CA C → → （3） = = 为常数）
m[fr-[m)=f(w为正整数) f(x) lim f(x) (3)lim x->xo A (B≠0) x→g(x)limg(x) B ✉囧
0 0 lim ( ) lim ( ) ( n n n x x x x f x f x A n → → = = 为正整数） 0 0 0 lim ( ) ( ) (3) lim ( 0) ( ) lim ( ) x x x x x x f x f x A B g x g x B → → → = =
例11im(3x2-2x+6) x→2 解：lim(3x2-2x+6)=3limx]-2limx+lim6 x→2 x→2 =3×22-2×2+6=14 米 ✉合
2 2 1 lim(3 2 6) x x x → 例 − + 2 2 2 2 2 2 lim(3 2 6) 3[lim ] 2lim lim6 x x x x x x x x → → → → 解： − + = − + 2 = − + = 3 2 2 2 6 14
例2求lim x3-1 x-3x+5 lim(x2-3x+5)limx2-lim3x+lim5 x->2 x→2 x→2 x→2 =(lim x)2-3lim x+lim5 x->2 x→2 x→2 =22-3.2+5=3≠0， x3-1 lim x3-lim 1 ∴.lim x→2 x→2 x2x2-3x+5 lim(x2-3x+5) 23-1 7 3 3
例 2 . 3 5 1 lim 2 3 2 − + − → x x x x 求 解 lim( 3 5) 2 2 − + → x x x lim lim 3 lim 5 2 2 2 → 2 → → = − + x x x x x (lim ) 3lim lim5 2 2 2 →2 → → = − + x x x x x 2 3 2 5 2 = − + = 3 0 , 3 5 1 lim 2 3 2 − + − → x x x x lim ( 3 5 ) lim lim 1 2 2 2 3 2 − + − = → → → x x x x x x . 37 = 3 2 1 3 − =