Question

洛比达法则无穷比无穷型证明

洛比达法则无穷比无穷型证明利用零比零型证明出无穷比无穷型

Answer 1

设f(x),g(x)->∞，则1/f(x),1/g(x)->0
limf(x)/g(x)=lim[1/g(x)]/[1/f(x)]
=lim[-g'(x)/g(x)^2]/[-f'(x)/f(x)^2]
=lim[f(x)^2/g(x)^2]*[g'(x)/f'(x)]
=[limf(x)/g(x)]^2*limg'(x)/f'(x)
所以limf(x)/g(x)=limf'(x)/g'(x)