Let $f:ℝ→ℝ$ be rael analtic function. Asume that $f$ is of finite order $1$ (An entire function is said to be of finite order if there exist numbers $a,r>0$ such that $$|f(x)|≤exp(|x|^{a})$$ for all $|x|>r$. The infimum of all numbers $a$ for which this inequality holds is called the function order of $f$).
Can we deduce that $lim_{x→+∞}f(x)=±∞$ or $lim_{x→-∞}f(x)=±∞$?
$f(x)=e^x$ and $g(x)=e^{-x}$ are each of order 1, but each fails one of your two limit tests.
On the other hand, if you intended to have a single test (that one of the two limits is $\pm \infty)$, then if a function fails that test, it must be bounded in absolute value (over the entire domain) by some constant, and hence of order 0.