Chapter2 第二节极限的运算 、极限的运算法则 二、两个重要极限
第二节 极限的运算 Chapter2 一、极限的运算法则 二、两个重要极限
、极限的运算法贝 定理设limf(x)=A,limg(x)=B,则 x→x x→x (1)lim[∫(x)±g(x)=limf(x)+limg(x)=A±B x→x x→x0 x→x (2) lim cf()=c lim f(x)=CA x-x x→x (3)im[f(x)·g(x)= x→x0 皿mf(x)img(x)=4:B (4) lim (r) lim f(x) (其中B≠0) x→+xg(x)limg(x)B x-x 注意定理的使用条件 Economic-mathematics 16-2 Wednesday, February 24, 2021
Economic-mathematics 16 - 2 Wednesday, February 24, 2021 定理 ( 0) lim ( ) lim ( ) ( ) ( ) (4) lim (3) lim [ ( ) ( )] lim ( ) lim ( ) (2) lim [ ( )] lim ( ) (1) lim [ ( ) ( )] lim ( ) lim ( ) 0 0 0 0 0 0 0 0 0 0 0 = = = = = = = + = → → → → → → → → → → → B B A g x f x g x f x f x g x f x g x A B c f x c f x cA f x g x f x g x A B x x x x x x x x x x x x x x x x x x x x x x 其 中 一、极限的运算法则 设 lim ( ) , lim ( ) ,则 0 0 f x A g x B x x x x = = → → 注意定理的使用条件
例1求lim(x2-5x+3) 解lim(x2-5x+3)=limx2-lim5x+3=1-5+3=-1. 解114x2d 例2求im x+5 3x+5) x→>2 imx2-lim3x+im5=22-3.2+5=3≠0, 2 lim x-limI ∴hi 23-17 x少2x2-3x+5im(x2-3x+5) 3 3 x→2 Economic-mathematics 16-3 Wednesday, February 24, 2021
Economic-mathematics 16 - 3 Wednesday, February 24, 2021 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 2 3 2 5 2 = − + = 3 0, 3 5 1 lim 2 3 2 − + − → x x x x lim( 3 5) lim lim1 2 2 2 3 2 − + − = → → → x x x x x x . 3 7 = 3 2 1 3 − = 例１ 求 lim( 5 3) 2 1 − + → x x x ． lim( 5 3) 2 1 − + → x x x lim lim 5 3 1 5 3 1. 1 2 1 = − + = − + = − → → x x x x 解
小结:1.设f(x)=anx2+a1xn1+…+an,则有 lim f(r)=ao(lim x)"+a,lim x)+.+a, =a0x0+a1x0+…+an=∫(x0) 2设/秒P(x),且(x0)≠0,则有 e(r) lim f(x) lim P(x) P(yol=/(o). x→x x→x0o im e(x) O(%o 若Q(x0)=0,则商的法则不能应用 Economic-mathematics 16-4 Wednesday, February 24, 2021
Economic-mathematics 16 - 4 Wednesday, February 24, 2021 小结: 1.设 f (x) = a0 x n + a1 x n−1 ++ an ,则有 n n x x n x x x x f x = a x + a x + + a − → → → lim ( ) 0 ( lim ) 1 ( lim ) 1 0 0 0 n n n = a x + a x + + a − 1 0 0 1 0 ( ). x0 = f 设 ,且 ( ) 0, 则 有 ( ) ( ) 2. ( ) = Q x0 Q x P x f x lim ( ) lim ( ) lim ( ) 0 0 0 Q x P x f x x x x x x x → → → = ( ) ( ) 0 0 Q x P x = ( ). x0 = f ( ) 0, . 若Q x0 = 则商的法则不能应用
例3求lm x2-1 x→1x2+2x-3 解x→1时,分子,分母的极限都是零 型) 0 先约去不为零的无穷小因子x-1后再求极限 lim (x+1)(x-1)(消去零因子法) x→小1x2+2x-3x1(x+3)(x x+1 m x→1x+32 Economic-mathematics 16-5 Wednesday, February 24, 2021
Economic-mathematics 16 - 5 Wednesday, February 24, 2021 解 例3 . 2 3 1 lim 2 2 1 + − − → x x x x 求 x → 1时,分子,分母的极限都是零. 先约去不为零的无穷小因子x −1后再求极限. ( 3)( 1) ( 1)( 1) lim 2 3 1 lim 1 2 2 1 + − + − = + − − → → x x x x x x x x x 3 1 lim 1 + + = → x x x . 2 1 = ) 0 0 ( 型 (消去零因子法)
2x3+3x2+5 例4求lm (—型) x→)∞7x3+4x2-1 解先用x3去除分子分母,分出无穷小,再求极限 35 2x3+3x2+5 2++ lim 2.3 2 Im x→0 7x+4x2-1 x→0 7+ 7 当a0≠0,b0≠0,m和n为非负整数时有 当n=m x+a1x+…+a lim x→b2x+b1x-1+…+bn 0,当n>m, 当 n< m Economic-mathematics 16-6 Wednesday, February 24, 2021
Economic-mathematics 16 - 6 Wednesday, February 24, 2021 例4 . 7 4 1 2 3 5 lim 3 2 3 2 + − + + → x x x x x 求 解 ( 型 ) , , . 先用x 3去除分子分母 分出无穷小 再求极限 3 3 3 2 3 2 4 1 7 3 5 2 lim 7 4 1 2 3 5 lim x x x x x x x x x x + − + + = + − + + → → . 7 2 = 当a0 0,b0 0,m和n为非负整数时有 = + + + + + + − − → n n n m m m x b x b x b a x a x a 1 0 1 1 0 1 lim , , 0 0 n m b a 当 = 0,当n m, ,当n m
x2-1 例5求皿3-x-1+x 解 √3-x-√1+x lim (x2-1)(√3-x+√1+x) 2(1-x) (+』)(3一X+^I+不) 2、2 Economic-mathematics 16-7 Wednesday, February 24, 2021
Economic-mathematics 16 - 7 Wednesday, February 24, 2021 . 3 1 1 lim 2 1 x x x x − − + − → 求 2(1 ) ( 1)( 3 1 ) 2 1 x x x x x − − − + + = → lim lim x x x x − − + − → 3 1 1 2 1 2 ( 1)( 3 1 ) lim 1 x x x x − + − + + = → 解 例5 = -2 2
二、两个重要极限 由表格和图形可见: sInd 当x→0时 sInd 1 lim x→>0y 土z 元 兀土 2 16 32 64 128 SInx 0.63660.90030.97450.99390.99840.99960.9999 x N取点少时 !取点多时 0.4 0.2 Economic-mathematics 16-8 Wednesday, February 24, 2021
Economic-mathematics 16 - 8 Wednesday, February 24, 2021 二、两个重要极限 1 sin 1. lim 0 = → x x x -10 -5 5 10 -0.2 0.2 0.4 0.6 0.8 1 -30 -20 -10 10 20 30 x -0.2 -0.1 0.1 0.2 0.3 y x x sin x 2 0.6366 4 8 16 32 64 128 0.9003 0.9745 0.9939 0.9984 0.9996 0.9999 取点少时 取点多时 1. sin → 0 → x x 当x 时， 由表格和图形可见：
二、两个重要极限 sInd lim x→>0y 注意两点: 00 型未定式 2 sIn 的结构式 相同 Economic-mathematics 16-9 Wednesday, February 24, 2021
Economic-mathematics 16 - 9 Wednesday, February 24, 2021 二、两个重要极限 1 sin 1. lim 0 = → x x x 1. 型未定式 “ ” 0 0 2. 的结构式 sin 相同 注意两点：
tanx 0 例6求lim or 0 解im tanx = lim sinx 1 →>0x r cos =lim sinr lim 1×1=1 x→>0Xx→>0coSx 请问:lim lim x→0tanx 5x→>0SinX Economic-mathematics 16-10 Wednesday, February 24, 2021
Economic-mathematics 16 - 10 Wednesday, February 24, 2021 “ ” 0 0 求 x x x tan lim →0 . x x x tan lim →0 ) cos sin 1 lim( 0 x x x x = → . 1 1 1. cos 1 lim sin lim 0 0 = = = → x → x x x x 例6 解 ________; 0 ________; 0 sin lim tan lim = = → → x x x x x x 请问： 1 1