Asymptotic analysis of coefficients of ordinary generating functions with radius of convergence $1$ seems to always predict polynomial growth rate

Question

Wikipedia gives the following formula for obtaining asymptotic information about the coefficients of an ordinary generating function from information about the generating function itself: if the generating function $G(a_n;x)$ is such that $$G(a_n;x)={A(x)+B(x)(1-\frac xr)^{-\beta}\over x^\alpha}$$ for some $\alpha$ and $\beta$ where $r$ is the radius of convergence of $G(a_n;x)$ and $A(x)$ and $B(x)$ are either entire or analytic to a radius of convergence strictly greater than $r$, then $$a_n \sim \frac{B(r)n^{\beta-1}}{r^{n+\alpha}\Gamma(\beta)}.$$ Suppose that $r=1$. Won't this always give an estimate of the form $$a_n\sim c_1n^{c_2}$$ for some constants $c_1$ and $c_2$? Clearly this cannot be true, since the generating function of many sequences not asymptotic to a polynomial has radius of convergence $1$.