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The improper integral is assumed to have a finite limit. Fix $\lambda >1$ and put $I(U,V)={\int }_U^V(\beta (x)-x)x^{-2}dx$. Convergence implies $I(X,\lambda X)\to 0$ and $I(X/\lambda ,X)\to 0$ as $X\to \infty$. By monotonicity,andThusBoth outer bounds tend to one as $\lambda \downarrow 1$, giving $\boxed{\beta (X)/X\to 1}$. This monotone integral Tauberian lemma uses monotonicity to control the value at a point from integral averages on adjacent multiplicative intervals; convergence of the integral alone would not control arbitrary local spikes.