A clinical researcher evaluates a new drug treatment by testing $H_0: \mu = 0$ against $H_1: \mu > 0$. The test statistic $T$ is normally distributed with mean $0$ under $H_0$ and mean $\delta > 0$ under $H_1$. If the researcher enforces a strict Significance Level $\alpha$, which statement correctly describes the behavior of the Power Function $\beta(\delta) = P(T > z_{1-\alpha} | \mu = \delta)$ as the sample size $n \to \infty$ for a fixed effect $\delta > 0$?