Consider the following function $$ f(x)=\frac{e^{-x}}{x^r}. $$ Question: for what real values of $r$ is this in $C(\mathbb{R}^+)^*$, the dual to the Banach space of continuous bounded functions on $\mathbb{R}^+$. My guess is $r<1$.
Reasoning: As long as $f\in L^1(\mathbb{R}^+)$, then the linear mapping $$ L_f:C(\mathbb{R}^+)\rightarrow \mathbb{R},\;\;L_f(g)\equiv\int_0^\infty fg dx $$ is continuous since $$ \left|L_f(g)\right|\leq \sup_{\mathbb{R}^+}\left(|g|\right)\|f\|_{L^1(\mathbb{R}^+)}. $$
Context: I need to find the point spectrum of the adjoint of a linear operator defined on $C(\mathbb{R}^+)$. Functions similar to $f(x)$ come up. I need to determine which ones are in $C(\mathbb{R}^+)^*$.
References: I did look several places. In particular, here, here, and here. None of what I looked at gave me concrete enough information (for my level of understanding) to make sure what I am saying is true.
Simpler problem?: Perhaps it makes it simpler for certain people to think on the whole line and consider the function $$ f(x)=\frac{e^{-|x|}}{|x|^r} $$ on $C(\mathbb{R})^*$ instead of $C(\mathbb{R}^+)^*$.