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bleach tybw part 4 release date - Feedz API

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 In a study of language evolution, a linguist models the frequency of a particular phoneme's usage over time with the function \( f(t) = \frac{3t^2 - 2t + 1}{t^2 + 1} \). Determine the horizontal asymptote of \( f(t) \) as \( t \) approaches infinity.

📌 To find the horizontal asymptote of the function \( f(t) = \frac{3t^2 - 2t + 1}{t^2 + 1} \) as \( t \to \infty \), we compare the degrees of the polynomial in the numerator and the denominator. Both the numerator and denominator are degree 2 polynomials. The horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.

📌 The leading coefficient of the numerator \( 3t^2 \) is 3, and of the denominator \( t^2 \) is 1. Thus, the horizontal asymptote is \( y = \frac{3}{1} = 3 \).

📌 Therefore, the horizontal asymptote is \(\boxed{y = 3}\).

📌 A zoologist observes that the population \( P \) of a certain species in the Amazon can be modeled by the equation \( P = \frac{500}{1 + e^{-0.1(t-5)}} \), where \( t \) is the number of months since the study began. Calculate the population of the species as \( t \) approaches infinity.

📌 To determine the population \( P \) as \( t \to \infty \), we analyze the behavior of the logistic function:

📌 \[ P = \frac{500}{1 + e^{-0.1(t-5)}} \]

📌 As \( t \to \infty \), the term \( e^{-0.1(t-5)} \to 0 \). Thus, the denominator approaches 1:

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