用等价无穷小代换极限lim(x→1)(x^2-1)/lnx
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![用等价无穷小代换极限lim(x→1)(x^2-1)/lnx](/uploads/image/z/679802-50-2.jpg?t=%E7%94%A8%E7%AD%89%E4%BB%B7%E6%97%A0%E7%A9%B7%E5%B0%8F%E4%BB%A3%E6%8D%A2%E6%9E%81%E9%99%90lim%28x%E2%86%921%29%28x%5E2-1%29%2Flnx)
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