Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.5.59

Chapter 2, Problem 2.5.59

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.
$f (x) = 1 - \ln x$

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The function given is $ f(x) = 1 - \ln x $. This is a transcendental function involving the natural logarithm.

The natural logarithm $ \ln x $ is defined for $ x > 0 $. Therefore, the domain of $ f(x) $ is $ x > 0 $.

As $ x \to 0^+ $, $ \ln x \to -\infty $. Therefore, $ f(x) = 1 - \ln x \to 1 - (-\infty) = +\infty $.

As $ x \to \infty $, $ \ln x \to \infty $. Therefore, $ f(x) = 1 - \ln x \to 1 - \infty = -\infty $.

There is a vertical asymptote at $ x = 0 $ since $ f(x) \to +\infty $ as $ x \to 0^+ $. There is no horizontal asymptote as $ x \to \infty $ because $ f(x) \to -\infty $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are crucial for analyzing the end behavior of functions, particularly as the input approaches infinity or a specific value. Understanding limits allows us to determine how functions behave at extreme values, which is essential for sketching graphs and identifying asymptotes.

Transcendental Functions

Transcendental functions, such as logarithmic, exponential, and trigonometric functions, are not algebraic and cannot be expressed as roots of polynomial equations. The function in the question, f(x) = 1 - ln(x), is a logarithmic function, which has unique properties affecting its limits and end behavior. Recognizing the characteristics of transcendental functions is vital for accurately analyzing their behavior and sketching their graphs.

Asymptotes

Asymptotes are lines that a graph approaches but never touches, indicating the behavior of a function as it tends towards infinity or a specific value. There are vertical, horizontal, and oblique asymptotes, each providing insight into the function's end behavior. Identifying asymptotes is essential for sketching accurate graphs, as they help illustrate how the function behaves at extreme values and around points of discontinuity.

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Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. f(x)=1−lnxf\(\left\)(x\(\right\))=1-\(\ln\) x

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Key Concepts

Limits

Transcendental Functions

Asymptotes

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.
$f (x) = 1 - \ln x$