Thus, the probability is $oxed{\dfrac{42}{55}}$.Question: A bioinformatician models the expression level of a gene as a function $ E(t) = rac{t^2 + 4t + 3}{t + 1} $, where $ t $ represents time in hours after treatment. Simplify $ E(t) $ and determine $ \lim_{t o -1} E(t) $, if it exists.
["Simplifying Gene Expression Model and Evaluating the Limit", "When analyzing gene expression dynamics, understanding how mathematical models behave near critical points—such as discontinuities or limits—can reveal important biological insights. Consider the expression level function defined by:", "$$\nE(t) = \frac{t^2 + 4t + 3}{t + 1}\n$$", "where $ t $ represents time in hours after treatment.", "---", "### Simplifying the Expression", "We begin by factoring the numerator:", "$$\nt^2 + 4t + 3 = (t + 1)(t + 3)\n$$", "Thus, the expression becomes:", "$$\nE(t) = \frac{(t + 1)(t + 3)}{t + 1}\n$$", "For all $ t <br/>\neq -1 $, we can cancel the common factor $ t + 1 $, yielding:", "$$\nE(t) = t + 3, \quad \ ext{for } t <br/>\neq -1\n$$", "This simplification shows that $ E(t) $ simplifies to a linear function except at $ t = -1 $, where the original expression is undefined due to division by zero.", "---", "### Analyzing the Limit as $ t \ o -1 $", "Although $ E(t) $ is undefined at $ t = -1 $, we compute the limit:", "$$\n\lim_{t \ o -1} E(t) = \lim_{t \ o -1} (t + 3) = -1 + 3 = 2\n$$", "Even though the function has a removable discontinuity at $ t = -1 $, the limit exists and equals 2. This suggests that while the observed expression level becomes undefined at exactly $ t = -1 $ hours (possibly indicating a critical time point), the long-term behavior stabilizes smoothly.", "---", "### Practical Implication for Bioinformatics", "In gene expression modeling, such limits help bioinformaticians interpret data around unstable time points. Here, despite the function blow-up at $ t = -1 $, the simplified linear form $ E(t) = t + 3 $ allows reliable extrapolation and analysis of gene activity immediately before and after this point.", "Thus, the probability or modeled expression at a near-zero time limit reduces cleanly to:", "$$\n\boxed{\dfrac{42}{55}} \quad \ ext{is incorrect in context; the correct limit is } \lim_{t \ o -1} E(t) = 2\n$$", "Wait — $\boxed{\dfrac{42}{55}}$ was incorrectly boxed. The limit is a numerical value, not a fraction from prior context.", "Correct final answer:", "$$\n\lim_{t \ o -1} E(t) = 2\n$$", "---", "Summary:\n- $ E(t) = t + 3 $ for $ t <br/>\ne -1 $\n- The limit as $ t \ o -1 $ exists and equals $ 2 $\n- Though undefined at $ t = -1 $, the function has a removable discontinuity", "This simplification enables accurate modeling of gene expression dynamics near transient biological effects, enhancing predictive accuracy in bioinformatic analyses."]
