用等价无穷小代换极限lim(x→1)(x^2-1)/lnx

用等价无穷小代换极限lim(x→1)(x^2-1)/lnx
用等价无穷小代换极限
lim(x→1)(x^2-1)/lnx

用等价无穷小代换极限lim(x→1)(x^2-1)/lnx
设lnx=y,x=e^y且x→1等于y→0.
lim(x→1)(x^2-1)/lnx
=lim(y→0)(e^y^2-1)/y
=lim(y→0)[e^(2y)-1]/y
设2y=n,y=n/2且y→0等于n→0.
=lim(n→0)(e^n-1)/(n/2)
=lim(n→0)ln(1+n)/(n/2) 等价无穷小代换
=lim(n→0)ln(1+n)^[(1/n)*2]
=lim(n→0)lne^2
=2