Understanding The Behavior Of F(x) = (2x) / (1 - X^2) As X Approaches Infinity

Jul 16, 2025 by Jeany 79 views

Which Statement Describes the Behavior of the Function f(x) = (2x) / (1 - x^2)?

This article delves into the behavior of the function f(x) = (2x) / (1 - x^2) as x approaches infinity. We will analyze the function's structure, identify its asymptotes, and ultimately determine the correct statement that describes its behavior. Understanding the behavior of functions, especially as their input values grow without bound, is a fundamental concept in calculus and mathematical analysis. This exploration will not only provide the answer to the question but also enhance your understanding of limits, asymptotes, and function analysis.

Understanding the Function f(x) = (2x) / (1 - x^2)

To accurately describe the behavior of the function f(x) = (2x) / (1 - x^2), a thorough analysis of its components and structure is essential. This function is a rational function, which means it is a ratio of two polynomials. The numerator is the linear expression 2x, and the denominator is the quadratic expression 1 - x^2. Understanding the degrees of these polynomials and their leading coefficients is crucial for determining the function's end behavior, particularly as x approaches infinity.

Analyzing the Numerator and Denominator: The numerator, 2x, is a linear function with a degree of 1. Its leading coefficient is 2. The denominator, 1 - x^2, is a quadratic function with a degree of 2. It's important to note that the leading coefficient of the denominator is -1 (the coefficient of the x^2 term). This negative leading coefficient will play a significant role in the function's behavior as x becomes very large, either positively or negatively. Furthermore, we can factor the denominator as (1 - x)(1 + x). This factorization helps us identify the vertical asymptotes of the function, which occur where the denominator equals zero. Setting (1 - x)(1 + x) = 0 gives us x = 1 and x = -1 as the vertical asymptotes.

Identifying Asymptotes: Asymptotes are lines that a function approaches as its input (x) or output (f(x)) approaches infinity. Vertical asymptotes occur where the denominator of a rational function equals zero, as we've already identified at x = 1 and x = -1. To determine the horizontal asymptote (the behavior as x approaches infinity), we compare the degrees of the numerator and the denominator. Since the degree of the denominator (2) is greater than the degree of the numerator (1), the horizontal asymptote is y = 0. This means that as x becomes very large (either positive or negative), the function's values will approach zero. The presence of these asymptotes significantly shapes the graph and the overall behavior of the function. We will further examine the implications of the horizontal asymptote in the subsequent sections.

Simplifying the Function (if possible): In some cases, rational functions can be simplified by canceling common factors between the numerator and the denominator. However, in this case, there are no common factors between 2x and (1 - x^2). Therefore, the function f(x) = (2x) / (1 - x^2) is already in its simplest form. Recognizing that the function cannot be simplified further allows us to focus on its asymptotic behavior and end behavior based on the degrees and leading coefficients of its numerator and denominator.

Determining the Limit as x Approaches Infinity

To determine the behavior of the function f(x) = (2x) / (1 - x^2) as x approaches infinity, we need to evaluate the limit of the function as x goes to both positive infinity and negative infinity. This involves analyzing how the function behaves when x takes on extremely large positive and negative values. The concept of limits is fundamental to calculus and provides a precise way to describe the behavior of functions as they approach certain values or infinity.

Evaluating the Limit as x Approaches Positive Infinity: When we consider x approaching positive infinity, we are interested in what happens to the function's values as x becomes increasingly large and positive. To evaluate the limit, we can divide both the numerator and the denominator by the highest power of x present in the denominator, which in this case is x^2. Doing so gives us:

lim (x→∞) [ (2x) / (1 - x^2) ] = lim (x→∞) [ (2x/x^2) / (1/x^2 - x^2/x2) ] = lim (x→∞) [ (2/x) / (1/x^2 - 1) ]

As x approaches infinity, the terms 2/x and 1/x^2 both approach zero. Therefore, the limit becomes:

lim (x→∞) [ (0) / (0 - 1) ] = 0

This indicates that as x approaches positive infinity, the function f(x) approaches 0. This confirms our earlier observation about the horizontal asymptote at y = 0.

Evaluating the Limit as x Approaches Negative Infinity: Similarly, we need to consider the behavior of the function as x approaches negative infinity. This means we are looking at what happens to the function's values as x becomes increasingly large in the negative direction. We use the same technique of dividing the numerator and the denominator by x^2:

lim (x→-∞) [ (2x) / (1 - x^2) ] = lim (x→-∞) [ (2x/x^2) / (1/x^2 - x^2/x2) ] = lim (x→-∞) [ (2/x) / (1/x^2 - 1) ]

As x approaches negative infinity, the terms 2/x and 1/x^2 still approach zero. Thus, the limit is:

lim (x→-∞) [ (0) / (0 - 1) ] = 0

This tells us that as x approaches negative infinity, the function f(x) also approaches 0. This further reinforces the existence of the horizontal asymptote at y = 0. It's important to note that the function approaches 0 from different directions as x goes to positive and negative infinity, due to the sign of the function.

Interpreting the Limits: The fact that the limit of f(x) as x approaches both positive and negative infinity is 0 is a critical piece of information. It tells us that the graph of the function will get arbitrarily close to the x-axis (y = 0) as x moves further away from the origin in either direction. This understanding is crucial for selecting the correct statement describing the function's behavior and for sketching the graph of the function.

Analyzing the Given Statements

Now that we have a thorough understanding of the function f(x) = (2x) / (1 - x^2) and its behavior as x approaches infinity, we can analyze the provided statements and determine which one accurately describes its behavior. We have established that the function has a horizontal asymptote at y = 0 and that the limits as x approaches both positive and negative infinity are 0.

The given statements are:

The graph approaches -2 as x approaches infinity.
The graph approaches 0 as x approaches infinity.
The graph approaches 1 as x approaches infinity.

Evaluating Statement 1: The statement “The graph approaches -2 as x approaches infinity” is incorrect. Our analysis of the limits as x approaches positive and negative infinity clearly showed that the function approaches 0, not -2. This statement might arise from a misunderstanding of the function's asymptotic behavior or a miscalculation of the limit.

Evaluating Statement 2: The statement “The graph approaches 0 as x approaches infinity” is correct. This aligns perfectly with our calculated limits and our identification of the horizontal asymptote at y = 0. As x becomes very large, either positively or negatively, the function's values get closer and closer to 0. This is the key behavior we've identified through our analysis.

Evaluating Statement 3: The statement “The graph approaches 1 as x approaches infinity” is incorrect. Similar to the first statement, this does not match the function's behavior as we determined it. The function approaches 0, not 1, as x approaches infinity. This statement could stem from a mistake in the limit calculation or a misunderstanding of how the numerator and denominator interact as x grows large.

Conclusion: Based on our analysis, the only accurate statement is that the graph approaches 0 as x approaches infinity. This conclusion is supported by our evaluation of the limits and our understanding of the function's horizontal asymptote. Choosing the correct statement requires a solid understanding of limits, asymptotes, and the behavior of rational functions.

Conclusion: The Correct Statement

In conclusion, after a comprehensive analysis of the function f(x) = (2x) / (1 - x^2), we have determined that the correct statement describing its behavior is: The graph approaches 0 as x approaches infinity.

This conclusion is supported by the following key points:

Horizontal Asymptote: We identified a horizontal asymptote at y = 0, which means the function's values approach 0 as x approaches either positive or negative infinity.
Limit Evaluation: We calculated the limits of the function as x approaches both positive infinity and negative infinity. In both cases, the limit was 0.
Polynomial Degrees: The degree of the denominator (x^2) is greater than the degree of the numerator (x), which is a characteristic of rational functions that have a horizontal asymptote at y = 0.

Understanding the behavior of functions as their inputs approach infinity is crucial in calculus and mathematical analysis. This example demonstrates how to analyze a rational function, identify its asymptotes, and evaluate limits to accurately describe its behavior. By considering the degrees of the polynomials, the limits at infinity, and the presence of asymptotes, we can confidently determine the correct statement that describes the function's end behavior. This process not only answers the specific question but also strengthens the understanding of fundamental concepts in function analysis.