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closed as off topic by Fernando Muro, Emil Jeřábek, j.c., Andreas Blass, Alexandre Eremenko Nov 22 '12 at 15:05
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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 $f/g\to 1$. 

