求一道求极限的高数题，lim（x趋近于无穷）[(x+1)/（x-1）]∧x

显示全部楼层 · 2011-10-26 22:38:48

解法一：原式=lim(x->∞){[(1+2/(x-1))^((x-1)/2)]^[2x/(x-1)]}
={lim(x->∞)[(1+2/(x-1))^((x-1)/2)]}^{lim(x->∞)[2x/(x-1)]}
=e^{lim(x->∞)[2x/(x-1)]}(应用重要极限lim(z->∞)[(1+1/z)^z]=e)
=e^{lim(x->∞)[2/(1-1/x)]}
=e^[2/(1-0)]
=e2；...

千问 · 2011-10-26 22:38:48

为简便省去极限号分子分母同时除以x,原式=[(1+1/x)/(1-1/x)]^x=[(1+1/x)]^x/[(1-1/x)]^x=[(1+1/x)]^x乘以[(1-1/x)]^{-x}=e×e=e2...

千问 · 2011-10-26 22:38:48

lim(x→∞)[(x+1)/(x-1)]^x=lim(x→∞){[1+2/(x-1)]^[(x-1)/2]}^2*[1+2/(x-1)]=lim(t→∞)(1+1/t)^2t，其中t=(x-1)/2=e^2...

千问 · 2011-10-26 22:38:48

e^2...