Problem 12 For each function, a. describe... [FREE SOLUTION]

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Calculus Concepts

Donald R. LaTorre, John W. Kenelly, Sherry S. Biggers

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 1: Problem 12

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). $$ j(u)=3-6 \ln u \text { for } u>0 $$

Short Answer

Expert verified

As $ u \to \infty $, $ j(u) \to -\infty $; as $ u \to 0^+ $, $ j(u) \to \infty $. No horizontal asymptotes.

Step by step solution

Analyze the Function

The function is given as $ j(u) = 3 - 6 \ln u $, where $ u > 0 $. This function involves the natural logarithm, which increases without bound as $ u $ increases. Therefore, $ -6 \ln u $ will decrease without bound as $ u $ gets larger.

Verbal Description of End Behavior

As $ u \to \infty $, the term $ -6 \ln u \to -\infty $. Thus, $ j(u) = 3 - 6 \ln u \to -\infty $. This means as $ u $ gets very large, $ j(u) $ becomes very negative. Conversely, as $ u \to 0^+ $, $ \ln u \to -\infty $ and $ -6 \ln u \to \infty $. Thus, $ j(u) = 3 + \infty = \infty $.

Write Limit Notation

In limit notation, the end behavior can be expressed as: \[\lim_{{u \to \infty}} j(u) = -\infty\]\[\lim_{{u \to 0^+}} j(u) = \infty\]

Determine Horizontal Asymptotes

For horizontal asymptotes, we investigate the limits at positive and negative infinity. In this case, since $ \lim_{{u \to \infty}} j(u) = -\infty $, there is no horizontal asymptote for $ u \to \infty $. As $ j(u) \to \infty $ for $ u \to 0^+ $, again, there is no horizontal asymptote towards $ u \to 0^+ $. Thus, there are no horizontal asymptotes.

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Key Concepts

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

Limit Notation

Limit notation is a mathematical tool that helps us describe the behavior of a function as the input approaches a certain value. It is essential for understanding how functions behave at the extremes, either toward positive infinity or near zero. Let's break it down:

For the given function, as $ u $ (the input) becomes very large (heading towards infinity), the function $ j(u) = 3 - 6 \ln u $ decreases without bound. Thus, using limit notation, we express this behavior as $ \lim_{{u \to \infty}} j(u) = -\infty $.
Conversely, if $ u $ approaches zero from the positive side (denoted as $ u \to 0^+ $), $ \ln u $ decreases infinitely, causing $ -6 \ln u $ to become very large positive. Hence, $ j(u) $ shoots up to infinity, which we write as $ \lim_{{u \to 0^+}} j(u) = \infty $.

Limit notation thus effectively captures and communicates these end behaviors, aiding in our understanding of the function's tendencies.

Horizontal Asymptotes

A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as $ x $ (or another input variable) gets very large or very small. It indicates the value that the function is approaching but never actually reaches. Here's what you need to know:

Horizontal asymptotes are often determined by the limits of the function as the input approaches $ \infty $ or $ -\infty $.
For our function $ j(u) = 3 - 6\ln u $, we need to see if there are any constant values that $ j(u) $ approaches when $ u $ becomes extremely large or approaches 0 from the positive side.
Since $ \lim_{{u \to \infty}} j(u) = -\infty $ and $ \lim_{{u \to 0^+}} j(u) = \infty $, the function does not level off at a particular value in either direction. Therefore, there are no horizontal asymptotes for this function.

Horizontal asymptotes provide information about the long-term behavior of functions, though in this case, $ j(u) $ does not have any.

Logarithmic Functions

Logarithmic functions include the natural logarithm function, denoted as $ \ln u $, which is the inverse of the exponential function. These functions are common in various mathematical calculations and real-world applications. Here's what is crucial to understand:

The natural logarithm $ \ln u $ is undefined for non-positive $ u $, hence the condition $ u > 0 $ for our function.
As $ u $ increases, $ \ln u $ also increases, though at a decreasing rate. This slow growth characterizes logarithmic functions: they expand infinitely, but quite leisurely compared to polynomials or exponentials.
In our function $ j(u) = 3 - 6\ln u $, this slow increase in $ \ln u $ dictates the decrease in the entire function $ j(u) $, due to the multiplication by $-6$.

Understanding the nature and behavior of logarithmic functions helps in analyzing the end behavior and effects on the overall function dynamics.

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91影视

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). $$ j(u)=3-6 \ln u \text { for } u>0 $$

Short Answer

Step by step solution

Analyze the Function

Verbal Description of End Behavior

Write Limit Notation

Determine Horizontal Asymptotes

Key Concepts

Limit Notation

Horizontal Asymptotes

Logarithmic Functions

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