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Chapter 1: Problem 13

For each function, a. describe the end behavior verbally, b. write limit notation for the end behavior, and c. write the equations for any horizontal asymptote(s). $$ s(t)=\frac{52}{1+0.5 e^{-0.9 t}} $$

Short Answer

Expert verified
End behavior: approaches 52 as \( t \to \infty \), zero as \( t \to -\infty \); limits: \( \lim_{t \to \infty} s(t) = 52 \), \( \lim_{t \to -\infty} s(t) = 0 \); horizontal asymptotes at \( y = 52 \) and \( y = 0 \).

Step by step solution

01

Analyze the Function

The function given is \( s(t) = \frac{52}{1 + 0.5 e^{-0.9 t}} \). This is a rational function where the numerator is a constant (52) and the denominator depends on an exponential function \( 0.5 e^{-0.9 t} \). We will examine the behavior as \( t \to \infty \) and \( t \to -\infty \).
02

End Behavior as \( t \to \infty \)

As \( t \to \infty \), the term \( e^{-0.9 t} \) becomes very small, approaching zero. Thus, the denominator approaches \( 1 + 0.5 \times 0 \) which simplifies to 1. Therefore, the function approaches \( \frac{52}{1} = 52 \). This suggests that the function levels off at 52 for very large \( t \).
03

Limit Notation for \( t \to \infty \)

In limit notation, the behavior as \( t \to \infty \) is written as: \( \lim_{t \to \infty} s(t) = 52 \). This indicates that the function approaches 52 as \( t \) increases indefinitely.
04

End Behavior as \( t \to -\infty \)

As \( t \to -\infty \), the term \( e^{-0.9 t} \) becomes very large. Therefore, the denominator \( 1 + 0.5 e^{-0.9 t} \) is dominated by the \( 0.5 e^{-0.9 t} \) term, which grows very large, making \( s(t) \approx \frac{52}{0.5 e^{-0.9 t}} \). As \( e^{-0.9 t} \to \infty \), \( s(t) \to 0 \).
05

Limit Notation for \( t \to -\infty \)

For \( t \to -\infty \), we express the behavior as: \( \lim_{t \to -\infty} s(t) = 0 \). This indicates that the function value approaches zero as \( t \) decreases indefinitely.
06

Horizontal Asymptotes

The horizontal asymptotes of the function can be determined from the limits as \( t \to \infty \) and \( t \to -\infty \). As the function approaches 52 from the positive direction and zero from the negative direction, we have a horizontal asymptote at \( y = 52 \) for \( t \to \infty \) and \( y = 0 \) for \( t \to -\infty \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Asymptotes
Horizontal asymptotes occur in rational functions when the function approaches a specific y-value as the variable x approaches infinity or negative infinity. In the given function,\( s(t) = \frac{52}{1 + 0.5 e^{-0.9 t}} \), we find horizontal asymptotes by observing the behavior of the function as \( t \to \infty \) or \( t \to -\infty \).

When \( t \to \infty \), the value of \( e^{-0.9 t} \) approaches zero, making the denominator simplify to 1. Therefore,\( s(t) \) approaches 52. So, there is a horizontal asymptote at \( y = 52 \).

In contrast, as \( t \to -\infty \), \( e^{-0.9 t} \) becomes very large, causing \( s(t) \) to approach 0. Hence, another horizontal asymptote is at \( y = 0 \).

  • For \( t \to \infty \): Horizontal asymptote at \( y = 52 \)
  • For \( t \to -\infty \): Horizontal asymptote at \( y = 0 \)
Limit Notation
Limit notation helps express the behavior of functions as the input approaches a specific value. It is a concise way to describe end behavior and asymptotes.

For the function \( s(t) = \frac{52}{1 + 0.5 e^{-0.9 t}} \):

  • As \( t \to \infty \), we write \( \lim_{t \to \infty} s(t) = 52 \). This means that the output of the function approaches 52 as \( t \) becomes very large.
  • Conversely, as \( t \to -\infty \), the notation \( \lim_{t \to -\infty} s(t) = 0 \) indicates that the function approaches 0 as \( t \) decreases without bound.


This notation is crucial for understanding how functions behave at their limits and provide insights into where horizontal asymptotes might exist. It helps in predicting the long-term behavior of rational functions.
Rational Functions
Rational functions are fractions where both the numerator and the denominator are polynomials. However, in the function \( s(t) = \frac{52}{1 + 0.5 e^{-0.9 t}} \), the numerator is a constant, and the denominator includes an exponential function.

Understanding these functions involves:
  • Recognizing how changes in the variable affect both parts.
  • Determining the end behavior as \( t \to \infty \) and \( t \to -\infty \).


In this case, the exponential decay inside the denominator plays a critical role. It adjusts the behavior of \( s(t) \) as \( t \) changes. As exponential terms grow or shrink, they influence the function's approach to its horizontal asymptotes. This makes rational functions with exponentials quite unique, distinguishing them from basic polynomial ratios.

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