Deriving information about asymptotics from finitness of a limit

Question

Let $f_1,f_2:\mathbb{R}\setminus\{0\}\to \mathbb{R^+}$ be two $C^1$ functions and $\alpha:\mathbb{R}\setminus \{0\}\to \mathbb{R}$ be a function from a Zygmund class (in particular it is Holder for every exponent less than $1$). Assume that for some $\alpha_1,\alpha_2 \in (0,1)$ the following limits are finite and nonzero: $$\lim_{h\to 0+} f_1(h) h^{\alpha_1}$$ and $$ \lim_{h\to 0+} f_2(-h) |-h|^{\alpha_2}.$$ Suppose I know that $$ \lim_{h\to 0+} ( f_1(h) \alpha(h) - f_2(-h)\alpha(-h) ) $$ is finite.

1. Does it imply that $\alpha_1 = \alpha_2$?
2. What if I assume in addition that $\alpha$ extends to a continuous function (or even to a Zygmund function) on the whole line?
3. If none of the above works, what can I get from my assumptions and what would be the weakest natural conditions to get $\alpha_1 = \alpha_2$?