Question

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

Short answer

Answer: The end behavior of the given transcendental function is as follows: as x approaches positive infinity, the function approaches negative infinity; the limit as x approaches negative infinity does not exist. There is a vertical asymptote at x = 0.

Step-by-step solution

Limits as x approaches positive infinity

Evaluate the limit of the function as x approaches positive infinity: $$\lim_{x \to \infty} (1 - \ln x)$$ As x approaches infinity, the natural logarithm of x, ln(x), will also approach infinity. Thus, the limit will be $$1 - \infty = -\infty$$

Limits as x approaches negative infinity

Evaluate the limit of the function as x approaches negative infinity: $$\lim_{x \to -\infty} (1 - \ln x)$$ As x approaches negative infinity, the function becomes undefined because the natural logarithm of a negative number is not defined. Therefore, the limit as x approaches negative infinity does not exist for this function.

Finding vertical asymptotes

To find if there is a vertical asymptote, we need to analyze the function for any values of x that make it undefined. The natural logarithm of 0 is undefined, so we have a vertical asymptote at x = 0.

Simple Sketch

Now that we have analyzed the limits and asymptotes for the given function, we can provide a simple sketch of the associated graph. The graph has a vertical asymptote at x = 0 and approaches negative infinity as x approaches positive infinity.

Key concepts

Transcendental Functions

Transcendental functions are a fascinating category in mathematics. Unlike algebraic functions, which involve operations like addition, subtraction, multiplication, division, and root extraction, transcendental functions include elements that transcend these operations. Examples include exponential functions, logarithms, and trigonometric functions.

For instance, the function you are working with, \(f(x) = 1 - \ln x\), is a transcendental function primarily because it includes the natural logarithm, \(\ln x\). Transcendental functions often exhibit unique behaviors and characteristics, which can be evaluated by examining their limits, asymptotes, and more. Understanding these functions enriches your grasp of calculus and aids in various scientific and engineering applications.

Limits at Infinity

When examining the behavior of functions as \(x\) approaches infinity, the concept of limits at infinity becomes crucial. It helps us understand what happens to a function's values when \(x\) becomes very large or very negative.

In the context of \(f(x) = 1 - \ln x\), evaluating the limit as \(x\) approaches positive infinity involves observing the behavior of \(\ln x\). Since \(\ln x\) increases without bound, the function approaches \(-\infty\) as \(x\) approaches \(\infty\).

Exploring limits also involves considering what happens as \(x\) approaches negative infinity. However, for \(\ln x\), negative values of \(x\) are undefined, so it's important to note that the function does not tend to any value for negative infinity. Limits at infinity are a fundamental part of analyzing the overall end behavior of transcendental functions.

Vertical Asymptotes

Vertical asymptotes occur in functions where the output tends toward infinity as the input gets close to a particular value. Identifying these asymptotes is key to understanding the domain and graphing behavior of the function.

For \(f(x) = 1 - \ln x\), the natural logarithm function \(\ln x\) is undefined at \(x = 0\) because \(\ln 0\) is undefined. Hence, \(x = 0\) represents a vertical asymptote for this function.

Vertical asymptotes signal boundaries beyond which the function cannot go, and they impact the function's graph by causing it to shoot up or down infinitely near the line \(x = 0\). Being aware of asymptotes is particularly useful when graphing to predict where the graph may break or tend toward infinity.

Graph Sketching

Graph sketching is an invaluable tool in calculus, allowing you to visualize the features and behaviors of functions. When sketching a function like \(f(x) = 1 - \ln x\), you integrate all the insights about limits and asymptotes.

Begin by noting the key behaviors from your analysis:

• As \(x\) approaches infinity, \(f(x)\) heads toward \(-\infty\).
• There is a vertical asymptote at \(x = 0\).

When drawing, plot these characteristics carefully:

• Herald the asymptote and ensure your graph approaches infinity around \(x = 0\).
• Show the function declining as \(x\) gets larger to depict the trend towards \(-\infty\).

With these steps, your sketch captures the essence of the function's behavior visually, aiding in more in-depth understanding and future calculus problem-solving.