Question: A herpetologist studies the growth of a lizard population, modeled by $ P(x) = rac{100}{1 + 9e^{-0.5x}} $. Find the range of $ P(x) $ as $ x $ approaches infinity.

["Exploring the Population Dynamics of Lizards: Understanding the Range of the Growth Model", "When studying animal populations, mathematical modeling helps scientists predict how species evolve over time under various ecological conditions. One such powerful model used in ecology is the logistic growth function, which captures the natural constraints on population expansion. A well-known example is:", "[\nP(x) = \frac{100}{1 + 9e^{-0.5x}}\n]", "This equation describes the lizard population, where $ x $ typically represents time (in months or years), and $ P(x) $ is the estimated population size. Understanding the range of this function—especially as $ x \ o \infty $—provides critical insight into the species' long-term sustainability.", "### Analyzing the Growth Model", "The function follows the standard logistic form:", "[\nP(x) = \frac{L}{1 + Ce^{-kx}}\n]", "where:\n- $ L $ is the carrying capacity—the maximum population size the environment can sustain,\n- $ C $ is a constant related to initial conditions,\n- $ k $ controls the rate of growth.", "In our model:\n- $ L = 100 $\n- $ C = 9 $\n- $ k = 0.5 $", "As $ x $ increases, the term $ e^{-0.5x} $ approaches zero because the exponent becomes increasingly negative:", "[\n\lim_{x \ o \infty} e^{-0.5x} = 0\n]", "### Finding the Limit as $ x \ o \infty $", "Substituting the limiting behavior:", "[\n\lim_{x \ o \infty} P(x) = \frac{100}{1 + 9 \cdot 0} = \frac{100}{1 + 0} = 100\n]", "Thus, the population approaches 100 as time progresses. This is the upper bound—what ecologists refer to as the maximum sustainable population.", "### Behavior at $ x \ o 0 $ and Positive $ x $", "Before reaching adulthood, the population starts lower. At $ x = 0 $:", "[\nP(0) = \frac{100}{1 + 9e^{0}} = \frac{100}{1 + 9} = \frac{100}{10} = 10\n]", "As $ x $ increases from 0, $ P(x) $ grows smoothly and monotonically toward 100, without recrossing it (since the logistic curve is strictly increasing).", "### The Full Range of $ P(x) $", "Combining these observations, the function $ P(x) $ ranges between 10 and 100, exclusive of 100 until infinitely far in the future. Therefore:", "[\n\ ext{Range as } x \ o \infty \ ext{: } (10, 100]\n]", "At the limit, 100 is approached asymptotically but never exceeded—consistent with ecological principles where resource limitations prevent overshooting carrying capacity.", "### Why This Matters for Conservation", "Understanding the full range of $ P(x) $ helps conservationists:\n- Set realistic population targets,\n- Monitor whether the environment can support growing numbers,\n- Detect imbalances—such as insufficient $ C $ (initial population size) leading to a lower carrying capacity.", "### Conclusion", "The logistic model $ P(x) = \frac{100}{1 + 9e^{-0.5x}} $ elegantly captures the lizard population’s growth limited by environmental constraints. As $ x \ o \infty $, the population stabilizes at 100, and thus its range is:", "[\n\boxed{(10, 100]}\n]", "This range underscores both the potential and the boundary of sustainable population size in real-world ecosystems."]