求数学极限的问题

来源：百度知道 编辑：UC知道 时间：2024/05/12 12:57:58
lim[x的二分之三次乘以(根号(x+2)-2乘以根号(x+1)+根号x] 当x无限趋向无穷大时,求极限
答案是负的四分之一

请大家帮忙,写出详细过程,小弟在此谢过了

是这样吗？

x^(3/2)*((x+2)^(1/2)-2(x+1)^(1/2)+x^(1/2))
=x^(3/2)*((1/((x+2)^(1/2)+(x+1)^(1/2))-(1/((x+1)^(1/2)+x^(1/2)))
=x^(3/2)*(-(x+2)^(1/2)+x^(1/2))/(((x+2)^(1/2)+(x+1)^(1/2))*((x+1)^(1/2)+x^(1/2)))
=x*(1-((x+2)/x)^(1/2)/(((x+2)/x)^(1/2)+((x+1)/x)^(1/2))*((x+1)/x)^(1/2)+1))
所以：
lim x^(3/2)*((x+2)^(1/2)-2(x+1)^(1/2)+x^(1/2))
=(1/4) lim x*(1-((x+2)/x)^(1/2)
=(1/4) lim ((1-((x+2)/x)^(1/2))/(1/x)
=(1/4) lim ((1/