Let $f$ a be a strictly positive function defined in the positive reals. Additionally suppose that for any $\delta > 0$ we have, as $t \to \infty$, $$ e^{-t^{1+\delta}} \ll f(t) \ll e^{t^{1+\delta}}$$ where we have used "Hardy's notation": $f(t) \ll g(t) \Leftrightarrow f(t) = o(g(t))$.
It is true then that for any $u \geq 0$ the limit $$ \lim_{t \to \infty} \frac{f(u+t)}{f(t)} $$ exists and is strictly positive?
The exponential, power and logarithmic cases are true, that is what led me to pose the question.
Thanks in advance