Function Behavior: F(x) = 2x/(1-x^2) Explained

$$

Let's analyze the behavior of the function $f(x) = \frac{2x}{1-x^2}$ as $x$ approaches infinity.

Understanding the Function

First, it's crucial to understand the components of the function. We have a rational function where the numerator is $2x$ and the denominator is $1-x^2$. As $x$ becomes very large, the term $x^2$ in the denominator will dominate the constant term 1. Therefore, we can approximate the function's behavior for large $x$ by considering only the highest powers of $x$ in the numerator and denominator.

Analyzing the Limit as x Approaches Infinity

To determine the function's behavior as $x$ approaches infinity, we need to evaluate the limit:

$\lim_{x \to \infty} \frac{2x}{1-x^2}$

We can divide both the numerator and the denominator by the highest power of $x$ present in the denominator, which is $x^2$:

$\lim_{x \to \infty} \frac{\frac{2x}{x^2}}{\frac{1}{x^2}-\frac{x^2}{x^2}} = \lim_{x \to \infty} \frac{\frac{2}{x}}{\frac{1}{x^2}-1}$

As $x$ approaches infinity, $\frac{2}{x}$ approaches 0, and $\frac{1}{x^2}$ also approaches 0. Therefore, the limit becomes:

$\lim_{x \to \infty} \frac{0}{0-1} = \frac{0}{-1} = 0$

This indicates that as $x$ approaches infinity, the function $f(x)$ approaches 0.

Analyzing the Limit as x Approaches Negative Infinity

Similarly, we should consider the behavior as $x$ approaches negative infinity:

$\lim_{x \to -\infty} \frac{2x}{1-x^2}$

Again, divide both the numerator and the denominator by $x^2$:

$\lim_{x \to -\infty} \frac{\frac{2x}{x^2}}{\frac{1}{x^2}-\frac{x^2}{x^2}} = \lim_{x \to -\infty} \frac{\frac{2}{x}}{\frac{1}{x^2}-1}$

As $x$ approaches negative infinity, $\frac{2}{x}$ approaches 0, and $\frac{1}{x^2}$ approaches 0. Therefore, the limit becomes:

$\lim_{x \to -\infty} \frac{0}{0-1} = \frac{0}{-1} = 0$

This confirms that as $x$ approaches negative infinity, the function $f(x)$ also approaches 0.

Conclusion

Based on our analysis, the correct statement is:

The graph approaches 0 as $x$ approaches infinity.

This conclusion is derived from evaluating the limits of the function as $x$ goes to both positive and negative infinity. The function $f(x) = \frac{2x}{1-x^2}$ tends towards 0 in both cases.

Additional Insights

Symmetry

Notice that the function $f(x) = \frac{2x}{1-x^2}$ is an odd function because $f(-x) = -f(x)$. This can be shown as follows:

$f(-x) = \frac{2(-x)}{1-(-x)^2} = \frac{-2x}{1-x^2} = -\frac{2x}{1-x^2} = -f(x)$

This symmetry implies that the graph of the function is symmetric with respect to the origin. Knowing this can help in visualizing the function’s behavior.

Vertical Asymptotes

The function has vertical asymptotes where the denominator is equal to zero:

$1 - x^2 = 0 \implies x^2 = 1 \implies x = \pm 1$

Thus, there are vertical asymptotes at $x = 1$ and $x = -1$. As $x$ approaches 1 or -1, the function tends to infinity or negative infinity.

Behavior Near Vertical Asymptotes

As $x$ approaches 1 from the left (i.e., $x \to 1^-$), $f(x)$ approaches infinity:

$\lim_{x \to 1^-} \frac{2x}{1-x^2} = \infty$

As $x$ approaches 1 from the right (i.e., $x \to 1^+$), $f(x)$ approaches negative infinity:

$\lim_{x \to 1^+} \frac{2x}{1-x^2} = -\infty$

As $x$ approaches -1 from the left (i.e., $x \to -1^-$), $f(x)$ approaches infinity:

$\lim_{x \to -1^-} \frac{2x}{1-x^2} = \infty$

As $x$ approaches -1 from the right (i.e., $x \to -1^+$), $f(x)$ approaches negative infinity:

$\lim_{x \to -1^+} \frac{2x}{1-x^2} = -\infty$

Graph Sketch

Considering all these behaviors, we can sketch the graph of the function. It approaches 0 as $x$ goes to infinity or negative infinity, has vertical asymptotes at $x = 1$ and $x = -1$, and is symmetric with respect to the origin. This comprehensive analysis ensures a solid understanding of the function’s behavior.

Further Elaboration

To further solidify the understanding, let's delve a bit deeper into the mathematical reasoning and practical implications. We've established that the function $f(x) = \frac{2x}{1-x^2}$ approaches 0 as $x$ tends towards infinity. This behavior is primarily dictated by the dominance of the $x^2$ term in the denominator compared to the linear term $2x$ in the numerator when $x$ is sufficiently large. In essence, the denominator grows at a faster rate than the numerator, causing the overall value of the fraction to diminish and converge to zero.

Importance of Limits

The concept of limits is fundamental in calculus, providing a rigorous way to describe the behavior of functions as their inputs approach specific values, whether finite or infinite. In this context, understanding limits helps us predict the long-term behavior of the function and identify any asymptotic tendencies. For instance, the horizontal asymptote at $y = 0$ (as $x$ approaches infinity) indicates that the function gets arbitrarily close to the x-axis but never actually touches it.

Practical Implications

In practical applications, functions like $f(x) = \frac{2x}{1-x^2}$ can model various phenomena in physics, engineering, and economics. For example, it could represent the response of a system to an external stimulus, the concentration of a substance in a chemical reaction, or the market share of a product as a function of time. Understanding the asymptotic behavior of such functions is crucial for making predictions and optimizing system parameters.

Alternative Approaches

While dividing by the highest power of $x$ is a common technique for evaluating limits at infinity, there are alternative approaches that can provide additional insights. For instance, L'Hôpital's Rule can be applied if the limit results in an indeterminate form such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$. However, in this case, applying L'Hôpital's Rule would require differentiation, which might not be necessary given the straightforward algebraic approach we've already used.

Common Mistakes

When analyzing the behavior of rational functions, it's important to avoid common pitfalls such as ignoring the signs of the terms or misinterpreting the dominance of terms. For example, mistakenly assuming that the function approaches 2 (based on the coefficient of the numerator) would lead to an incorrect conclusion. Careful attention to detail and a systematic approach are essential for accurate analysis.

In summary, the function $f(x) = \frac{2x}{1-x^2}$ provides a rich context for exploring concepts such as limits, asymptotes, symmetry, and practical applications. By understanding its behavior as $x$ approaches infinity, we gain valuable insights into the broader landscape of mathematical analysis and its relevance to real-world phenomena. The correct statement is that the graph approaches 0 as $x$ approaches infinity, a conclusion supported by both algebraic manipulation and a conceptual understanding of limits and asymptotic behavior.