Solution: As $ x o \infty $, $ e^{-0.5x} o 0 $, so $ P(x) o rac{100}{1 + 0} = 100 $. As $ x o -\infty $, $ e^{-0.5x} o \infty $, so $ P(x) o 0 $. The range is $ (0, 100) $. oxed{(0, 100)}
["# Understanding the Asymptotic Behavior of Probability Functions: A Closer Look at $ P(x) \ o (0, 100) $", "When analyzing probability density functions (PDFs) in mathematical modeling, especially those involving exponential decay like $ e^{-0.5x} $, understanding limits and long-term behavior is crucial. One key observation is the asymptotic convergence of such functions, illustrated here by the formula:", "$$\n\lim_{x \ o \infty} e^{-0.5x} = 0 \quad \Rightarrow \quad P(x) \ o \frac{100}{1 + 0} = 100\n$$\n$$\n\lim_{x \ o -\infty} e^{-0.5x} = \infty \quad \Rightarrow \quad P(x) \ o 0\n$$", "### Why Does $ P(x) \ o 100 $ as $ x \ o \infty $?", "As $ x $ grows large and positive, the exponential term $ e^{-0.5x} $ rapidly diminishes toward zero. This decay mimics processes in nature and finance where values approach a maximum bound — for example, diminishing returns or saturation effects. Multiplying this by 100 scales the result to a meaningful upper limit, ensuring $ P(x) $ approaches exactly 100 but never reaches or exceeds it.", "### Why Does $ P(x) \ o 0 $ as $ x \ o -\infty $?", "Conversely, as $ x $ approaches negative infinity (i.e., moving toward extreme values where $ -0.5x $ becomes very large positive), $ e^{-0.5x} $ grows without bound. Yet, normalized by an increasing denominator like $ 1 + e^{-0.5x} $, the ratio $ P(x) $ collapses toward zero. This reflects how probabilities must remain bounded between 0 and 1, capturing uncertainty that diminishes to hyper-negligible values in extreme negative domains.", "### The Full Range of $ P(x) $", "Combining these asymptotic behaviors, we find the range of $ P(x) $ is strictly the open interval:", "$$\n(0, 100)\n$$", "This means $ P(x) $ values get arbitrarily close to 0 or 100 but never touch these bounds. Such a function effectively models bounded probability processes, commonly used in genetics, finance, and decay phenomena—where outcomes settle into definable limits without overshooting or reaching extremes.", "### Final Thoughts", "The behavior $ \lim_{x \ o \infty} P(x) = 100 $, $ \lim_{x \ o -\infty} P(x) = 0 $, and the confirmed range $ (0, 100) $ reveal the smooth, bounded nature of exponential decay models. Recognizing these limits is essential for accurate interpretation and application in real-world statistical and scientific contexts.", "---", "Boxed Result:\n$$\n\ ext{Range of } P(x) = (0, 100)\n$$", "---", "This refined explanation enhances SEO by integrating relevant keywords like “asymptotic behavior,” “exponential decay limits,” “probability range,” and “normalized probability,” while remaining clear and accessible for readers interested in mathematical modeling and statistical functions."]
