Bounded ration of functions If $f(x)$ is a continuous function on $\mathbb R$, and $f$ is doesn't vanish on $\mathbb R$, does this imply that the function $\frac{f\,'(x)}{f(x)}$ be bounded on $\mathbb R$? The function $1/f(x)$ will be bounded because $f$ doesn't vanish, and I guess that the derivative will reduce the growth of the function $f$, so that the ration will be bounded, is this explanation correct!?

Take $f(x)=e^{x^2}$ then $$\frac{f'(x)}{f(x)}= 2x$$ which is not bounded on $\mathbb{R}$

Also if $f(x)$ does not vanish it does not mean that $\frac{1}{f(x)}$ is bounded for example take $e^{-x}$.