常用函数的麦克劳林公式 5 SIn=d +(-1)+2n+1 +0(x 2n+2 3!5 (2n+1) cOS=1中 +…+(-1) 2 +O(x2) 2!4!6 (2n)! H+1 ln(1+x)=x-+ +(-1) +0(x 23 n+1 =1+x+x2+…+x"+0(x") m(m (1+x)"=1+m+ x-+ 2 m(m-1)…(m-n+1) x"+0(x")
常用函数的麦克劳林公式 ( ) (2 1)! ( 1) 3! 5! sin 2 2 3 5 2 1 + + + + = − + − + − n n n o x n x x x x x ( ) (2 )! ( 1) 2! 4! 6! cos 1 2 2 4 6 2 n n n o x n x x x x x = − + − ++ − + ( ) 1 ( 1) 2 3 ln(1 ) 1 2 3 1 + + + + + = − + − + − n n n o x n x x x x x 1 ( ) 1 1 2 n n x x x o x x = + + + + + − ( ) ! ( 1) ( 1) 2! ( 1) (1 ) 1 2 n n m x o x n m m m n x m m x m x + − − + + + − + = + +
e+2cosx-3 例2计算lim x→>0 4 解∵ex=1+x2+x+0(x4) Cosr=1-x x4 +0(x) e+2c0sx-3=(+2·,)x+0(x) x+0(x 原式=lim 12 7 x->0 12
例 2 计算 4 0 2cos 3 lim 2 x e x x x + − → . 解 ( ) 2! 1 1 2 4 4 2 e x x o x x = + + + ( ) 2! 4! cos 1 5 2 4 o x x x x = − + + ) ( ) 4! 1 2 2! 1 2cos 3 ( 4 4 2 e x x o x x + − = + + 12 7 ( ) 12 7 lim 4 4 4 0 = + = → x x o x x 原式