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closed as off topic by Fernando Muro, Emil Jeřábek, j.c., Andreas Blass, Alexandre Eremenko Nov 22 '12 at 15:05Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question. 


No. Take $f(x)=\exp(1/x),\; g(x)=\exp(1/x^2)$. Then $\lim(f'/f)/(g'/g)=\lim(\log f)'/(\log g)'=\infty$, but $f/g\to 1$. 

