$f(x)\geq e^x$ implies $xf'(x)/f(x)\to \infty$

Question

Is it true that $f(x)\geq e^x$ for large $x$ implies $xf'(x)/f(x)\to \infty$? We assume that $f$ is smooth.

Answer 1

I'll show $\limsup_{x \to \infty} \frac{xf'(x)}{f(x)} = +\infty$. If we let $g(x) = \frac{f(x)}{e^x}$, then $g(x) \ge 1$ and $x\frac{f'(x)}{f(x)} = x(1+\frac{g'(x)}{g(x)})$ for each $x$. If the limsup of this were not $+\infty$, then $g'(x)$ must be less than $-g(x) \le -1$ for all large $x$. But this contradicts $g(x) \ge 1$ for all large $x$.