求 lim(x→∞)[1-(3/x)]^x的极限.

来源：学生作业帮助网编辑：作业帮时间：2024/05/06 16:34:50

求 lim(x→∞)[1-(3/x)]^x的极限.
求 lim(x→∞)[1-(3/x)]^x的极限.

求 lim(x→∞)[1-(3/x)]^x的极限.
lim(x→∞)[1-(3/x)]^x
=lim(x→∞)e^{ln[1-(3/x)]^x}
=e^lim(x→∞)xln[1-(3/x)]
lim(x→∞)xln[1-(3/x)]
=lim(x→∞){ln[1-(3/x)]}/(1/x)
由洛比达法则
=lim(x→∞)[x/(x-3)]*(-3/-x^2)/[1/(-x^2)]
=lim(x→∞)[x/(x-3)])]*-3
=-3
所以lim(x→∞)[1-(3/x)]^x=e^lim(x→∞)xln[1-(3/x)]=e^（-3）
本来很简单的题.在电脑上写出来看起来很复杂.Orz
用笔照这过程写一遍应该知道了.

lim(x→∞)[1-(3/x)]^x
=lim(x→∞)[1-(3/x)]^(3x/3)
=lim(x→∞){[1-(3/x)]^(x/3)}^3
=1/e^3
因为lim(x→∞){[1-(3/x)]^(x/3)=1/e

lim[1-(3/x)]^x
=lim{[1+(-3/x)]^(-x/3)}^(-3)
因为lim[1+(1/t)]^t=e
=e^(-3)

lim(x→0)[1-(3/x)]^x
代换x=-3t x→0 t→0
=lim(t→0)(1+1/t)^(-3t)
=lim(t→0)[(1+1/t)^t]^(-3)
=e^(-3)

求lim(x→0)x cos 1/x lim(x→∞)x^2/ (3x-1)的极限求极限lim x→∞ (x/(1+x))^x lim (1-1/3x)^x x→∞ 求极限lim(1+1/x)^x=e 求极限 lim [(3+x)/(6+x)]^[(x-1)/2]= x→∞求极限 lim [(3+x)/(6+x)]^[(x-1)/2]=x→∞ 求极限lim(x→0)sinxsin(1/x);lim(x→∞)(arctanx/x) 求lim(√x^2+1)-x x→-∞ 求极限lim(x→∞)(2x/3-x)-(2/3x)求极限lim(x→5)根号(x-1)-2/x-5 求lim x→无穷大 (1-3/x)^x f(x)=(2+e^x)/1+e^2x)+ | x|sin1/x求 1)lim[x→+∞]f(x); 2)lim[x→-∞]f(x); 3)lim[x→∞]f(x) 求极限 lim x→∞ X方-1除以3X方-X-1 求极限：lim(x→∞)((x+1)/x)^3x会用word的最好求lim x→∞ (x+1/x)3x的平方,极限要过程求极限 lim(x→∞）（x+3/x-1)x次幂求极限lim(x→∞)[(2x+5)/(2x+3)]^(x+1) lim（ x→∞）（1+x / x)^3x 烦人的数学微积分1、lim(x→0)Sin3x/Sin2x2、lim(x→0)x/3*Sin3x3、lim(x→0)[Xsin(3/x)+(tanx/2x)]4、lim(x→3)Sin(x-3)/(X^2-X-6)5、lim(x→∞)[1+(1/2x)^x6、lim(x→∞)(x+3/x)^x7、lim(x→0)（1-x/2)^1/x+118、求下列函数的间断利用罗必达法则求极限lim x→∞x^n/e^ax(a>0,n为正整数)lim x→1 lnx/(x-1)lim (x^-3x^2+2)/(x^3-x^2+x+1) 高数试题利用落必达法则求极限lim x→∞x^n/e^ax(a>0,n为正整数)lim x→1 lnx/(x-1)lim (x^-3x^2+2)/(x^3-x^2+x+1)