Analyzing End Behavior Of F(x) = (7x^2 + X + 1) / (x^4 + 1) Using Limits

In mathematics, particularly in calculus and analysis, understanding the end behavior of a function is crucial for sketching graphs, analyzing long-term trends, and solving various real-world problems. The end behavior describes what happens to the function's output (${f(x)}$) as the input (${x}$) approaches positive or negative infinity. For rational functions, which are ratios of two polynomials, the end behavior is primarily determined by the degrees and leading coefficients of the polynomials in the numerator and the denominator.

To effectively analyze the end behavior, we employ the concept of limits. A limit describes the value that a function approaches as the input approaches a certain value, including infinity. By evaluating limits as ${x}$ approaches positive and negative infinity, we can precisely describe the function's trend at its extremes. This analysis is not just an abstract mathematical exercise; it has practical applications in fields like physics, engineering, and economics, where understanding long-term trends and stability is paramount.

In the subsequent sections, we will delve deeper into the specific example of the rational function ${f(x) = \frac{7x^2 + x + 1}{x^4 + 1}}$ to illustrate how limits are used to determine the end behavior. We will break down the function, identify key components, and step-by-step apply limit techniques to arrive at a conclusive understanding of its end behavior.

Analyzing the Given Function: f(x) = (7x^2 + x + 1) / (x^4 + 1)

The given function is a rational function defined as ${f(x) = \frac{7x^2 + x + 1}{x^4 + 1}}$. To analyze the end behavior of this function, we need to examine how the function behaves as ${x}$ approaches both positive infinity (${\infty}$) and negative infinity (${-\infty}$). The numerator of the function is the polynomial ${7x^2 + x + 1}$, and the denominator is the polynomial ${x^4 + 1}$.

The degree of a polynomial is the highest power of the variable in the polynomial. In the numerator, the highest power of ${x}$ is 2, so the degree of the numerator is 2. The leading coefficient of the numerator is the coefficient of the term with the highest power, which is 7. In the denominator, the highest power of ${x}$ is 4, so the degree of the denominator is 4. The leading coefficient of the denominator is 1.

The comparison of the degrees of the numerator and the denominator is crucial in determining the end behavior of the rational function. When the degree of the denominator is greater than the degree of the numerator, as in this case, the function will approach 0 as ${x}$ approaches infinity or negative infinity. This intuitive understanding will be rigorously demonstrated using limits in the following sections. Specifically, we will focus on how the higher powers of ${x}$ in the denominator will dominate the behavior of the function as ${x}$ becomes extremely large (positive or negative), causing the overall value of the function to tend towards zero. This principle is a fundamental aspect of rational function analysis and is essential for predicting long-term trends and stability in mathematical models.

Using Limits to Determine End Behavior

The most rigorous method for determining the end behavior of a function is by using limits. We need to evaluate the limits of ${f(x)}$ as ${x}$ approaches positive infinity and negative infinity. That is, we need to find:

${ \lim_{x \to \infty} \frac{7x^2 + x + 1}{x^4 + 1} }$

and

${ \lim_{x \to -\infty} \frac{7x^2 + x + 1}{x^4 + 1} }$

To evaluate these limits, we divide both the numerator and the denominator by the highest power of ${x}$ that appears in the denominator, which in this case is ${x^4}$. This technique simplifies the expression and allows us to easily see the end behavior as ${x}$ approaches infinity. Dividing every term in the numerator and the denominator by ${x^4}$ gives us:

${ \lim_{x \to \infty} \frac{\frac{7x^2}{x^4} + \frac{x}{x^4} + \frac{1}{x^4}}{\frac{x^4}{x^4} + \frac{1}{x^4}} = \lim_{x \to \infty} \frac{\frac{7}{x^2} + \frac{1}{x^3} + \frac{1}{x^4}}{1 + \frac{1}{x^4}} }$

Now, as ${x}$ approaches infinity, terms of the form ${\frac{c}{x^n}}$ (where ${c}$ is a constant and ${n}$ is a positive integer) will approach 0. Thus, we have:

${ \lim_{x \to \infty} \frac{\frac{7}{x^2} + \frac{1}{x^3} + \frac{1}{x^4}}{1 + \frac{1}{x^4}} = \frac{0 + 0 + 0}{1 + 0} = \frac{0}{1} = 0 }$

Similarly, we evaluate the limit as ${x}$ approaches negative infinity:

${ \lim_{x \to -\infty} \frac{\frac{7x^2}{x^4} + \frac{x}{x^4} + \frac{1}{x^4}}{\frac{x^4}{x^4} + \frac{1}{x^4}} = \lim_{x \to -\infty} \frac{\frac{7}{x^2} + \frac{1}{x^3} + \frac{1}{x^4}}{1 + \frac{1}{x^4}} }$

As ${x}$ approaches negative infinity, terms of the form ${\frac{c}{x^n}}$ still approach 0 (note that even if ${x}$ is negative, ${x^2}$ and ${x^4}$ are positive, and ${x^3}$ will approach 0 as well). Therefore:

${ \lim_{x \to -\infty} \frac{\frac{7}{x^2} + \frac{1}{x^3} + \frac{1}{x^4}}{1 + \frac{1}{x^4}} = \frac{0 + 0 + 0}{1 + 0} = \frac{0}{1} = 0 }$

These limit evaluations demonstrate that as ${x}$ approaches both positive and negative infinity, the function ${f(x)}$ approaches 0. This confirms that the end behavior of the function is such that the function values get arbitrarily close to 0 as ${x}$ moves away from the origin along the x-axis.

Conclusion: End Behavior of f(x) and Its Implications

In summary, through the rigorous application of limits, we have shown that:

${ \lim_{x \to \infty} \frac{7x^2 + x + 1}{x^4 + 1} = 0 }$

and

${ \lim_{x \to -\infty} \frac{7x^2 + x + 1}{x^4 + 1} = 0 }$

This result definitively describes the end behavior of the function ${f(x) = \frac{7x^2 + x + 1}{x^4 + 1}}$. As ${x}$ approaches positive or negative infinity, the function values approach 0. Graphically, this means that the graph of the function will get closer and closer to the x-axis (the line ${y = 0}$) as ${x}$ moves further away from the origin in either direction.

This type of analysis is not merely a theoretical exercise; it has profound implications in various fields. For instance, in engineering, understanding the end behavior of a system can help determine its long-term stability. In economics, it can be used to model the long-term behavior of markets or economic indicators. In physics, it can describe the behavior of potentials or fields at large distances.

Moreover, understanding the end behavior is crucial in sketching the graph of a function. It provides essential information about the function's asymptotic behavior, helping to create a more accurate representation of the function's overall shape. In the case of our example, knowing that the function approaches 0 as ${x}$ goes to infinity helps us visualize the graph flattening out towards the x-axis.

In conclusion, the analysis of end behavior using limits is a powerful tool in understanding and predicting the behavior of functions. It provides critical insights into the long-term trends and stability of mathematical models, making it an indispensable concept in both theoretical mathematics and applied sciences.