Problem 283 - Calculus Volume 1

Question

For the following exercises, find the horizontal and vertical asymptotes. $$ f(x)=x-\sin x $$

Step-by-Step Solution

Answer

The function \( f(x) = x - \sin x \) has no horizontal or vertical asymptotes.

Step 1: Define horizontal asymptote

Horizontal asymptotes are determined by analyzing the behavior of a function as \( x \) approaches infinity or negative infinity. To find any horizontal asymptotes for \( f(x) = x - \sin x \), examine the behavior as \( x \to \infty \) and \( x \to -\infty \).As \( x \to \infty \), \( \sin x \) oscillates between -1 and 1, so it has negligible impact on the function \( f(x) = x \), which tends to \( \infty \). Thus, no horizontal asymptote exists in this case.Similarly, as \( x \to -\infty \), \( f(x) = x - \sin x \) behaves as \( x \) because \( \sin x \) remains negligible, tending towards \( -\infty \). So, there is no horizontal asymptote here either.

Step 2: Define vertical asymptote

Vertical asymptotes occur where the function becomes undefined or blows up to infinity. The function \( f(x) = x - \sin x \) is defined for all real numbers since it is a combination of a polynomial \( x \) and the sine function \( \sin x \), both of which are continuous everywhere. This means there are no values of \( x \) that result in division by zero or other undefined behavior. Therefore, \( f(x) \) has no vertical asymptotes.

Key Concepts

Understanding Horizontal Asymptotes

When we talk about horizontal asymptotes, we're discussing the behavior of a function as it moves towards very large positive or negative values of \( x \). Essentially, horizontal asymptotes indicate if there is a horizontal line that the graph of a function approaches but never quite touches.To determine this for the function \( f(x) = x - \sin x \), we examine the behavior as \( x \to \infty \) and \( x \to -\infty \):

• As \( x \to \infty \), the term \( \sin x \) oscillates within a small range between -1 and 1. However, this oscillation becomes negligible compared to the increasing value of \( x \). As a result, \( f(x) \) behaves like \( x \), tending towards \( \infty \).
• Similarly, as \( x \to -\infty \), \( f(x) \) resembles \( x \) because \( \sin x \) remains minor, leading to a behavior towards \( -\infty \).

Thus, without the graph approaching a constant but rather continuously increasing or decreasing, we conclude that there are no horizontal asymptotes for \( f(x) = x - \sin x \).

Exploring Vertical Asymptotes

Vertical asymptotes occur when certain functions become undefined or increase or decrease indefinitely at specific points. Typically, vertical asymptotes are found where a function involves division by zero or the function demonstrates undefined behavior at particular \( x \) values.For the function \( f(x) = x - \sin x \), it's helpful to note the characteristics of both components:

• The term \( x \) is linear and exists everywhere across real numbers without limitations.
• The sine function, \( \sin x \), also extends continuously over real numbers and does not lead to any undefined expressions.

Because neither \( x \) nor \( \sin x \) introduces any undefined zones or infinite spikes, there are no inputs for which \( f(x) \) becomes undefined. Hence, there are no vertical asymptotes for this function.

Function Behavior at Infinity

The concept of function behavior at infinity helps us understand how a function acts as \( x \) approaches very large positive or negative values. This helps in identifying some special properties or overall trends of the graph.In the case of \( f(x) = x - \sin x \):

• As \( x \to \infty \), the impact of \( \sin x \) dwindles due to its bounded oscillation, leading \( f(x) \) to pursue the straightforward linear growth of the \( x \) term towards \( \infty \).
• Inversely, as \( x \to -\infty \), the function similarly sees \( \sin x \) as minimal in comparison to the linear progression of \( x \), driving \( f(x) \) towards \( -\infty \).

Thus, the absence of real bounds or significant deviation highlights the straightforward linear nature of \( f(x) \) at extreme values, which is critical for understanding its end behavior.