Evaluate Limit At Infinity Lim (8x + 2) / (2x^2 - 5x + 9)

Introduction: Understanding Limits at Infinity

In the realm of calculus, evaluating limits is a fundamental concept, particularly when we delve into the behavior of functions as their input approaches infinity. The limit ${\lim _{x \rightarrow \infty} \frac{8 x+2}{2 x^2-5 x+9}}$ is a classic example of such a scenario, where we seek to understand the function's trend as x grows without bound. This exploration is not merely an academic exercise; it has profound implications in various fields, including physics, engineering, and economics, where understanding asymptotic behavior is crucial for modeling real-world phenomena. To evaluate this limit, we must employ techniques that allow us to dissect the function's structure and identify its dominant terms as x becomes infinitely large. These techniques often involve algebraic manipulations and a keen understanding of how different types of functions behave at extreme values. This article will provide a detailed, step-by-step guide on how to evaluate this limit, ensuring a clear understanding of the underlying principles and methodologies. We will begin by discussing the general approach to limits at infinity, then apply these principles to the specific function at hand, and finally, interpret the result in the broader context of calculus and its applications. This comprehensive approach aims to equip readers with the necessary tools to tackle similar problems and appreciate the elegance and power of limit evaluation in mathematical analysis. Furthermore, understanding limits at infinity is crucial for grasping concepts like asymptotes and end behavior of functions, which are essential for sketching graphs and analyzing functions in detail. So, let's embark on this journey to unravel the mysteries of limits at infinity and master the techniques required to solve them.

Step-by-Step Evaluation of ${\lim _{x \rightarrow \infty} \frac{8 x+2}{2 x^2-5 x+9}}$

To evaluate the limit ${\lim _{x \rightarrow \infty} \frac{8 x+2}{2 x^2-5 x+9}}$, we employ a standard technique for rational functions: dividing both the numerator and the denominator by the highest power of x present in the denominator. In this case, the highest power of x in the denominator is x². This step is crucial because it allows us to simplify the expression and reveal the function's behavior as x approaches infinity. By dividing each term by x², we effectively normalize the function, making it easier to analyze its asymptotic behavior. This method is based on the principle that the terms with the highest powers dominate the function's behavior as x becomes very large. The lower-order terms become insignificant in comparison, and dividing by the highest power helps to highlight the dominant terms. Therefore, dividing by x² is not just an algebraic manipulation; it is a strategic move that allows us to focus on the essential components of the function as x tends to infinity. This approach is widely applicable to rational functions and is a cornerstone of limit evaluation techniques. After performing this division, we will be able to clearly see which terms approach zero as x approaches infinity, and this will lead us to the final answer. The next steps will detail the actual division and simplification process, ensuring a clear understanding of each step involved in evaluating this limit. This method not only provides the answer but also enhances our understanding of how rational functions behave at extreme values.

Dividing both the numerator and the denominator by x², we get:

${ \lim _{x \rightarrow \infty} \frac{8 x+2}{2 x^2-5 x+9} = \lim _{x \rightarrow \infty} \frac{\frac{8 x}{x^2}+\frac{2}{x^2}}{\frac{2 x^2}{x^2}-\frac{5 x}{x^2}+\frac{9}{x^2}} }$

Simplifying the fractions, we obtain:

${ = \lim _{x \rightarrow \infty} \frac{\frac{8}{x}+\frac{2}{x^2}}{2-\frac{5}{x}+\frac{9}{x^2}} }$

Now, as x approaches infinity, the terms ${\frac{8}{x}}$, ${\frac{2}{x^2}}$, ${\frac{5}{x}}$, and ${\frac{9}{x^2}}$ all approach 0. This is because any constant divided by a quantity that grows without bound will tend towards zero. This observation is the key to evaluating the limit. It allows us to replace these terms with zero, simplifying the expression significantly. Understanding this behavior is crucial for handling limits at infinity. It highlights the dominance of constant terms over terms that tend to zero. This step is not just a mathematical trick; it is a direct consequence of the definition of a limit and the properties of infinity. By recognizing these terms' behavior, we can reduce the complex expression to a simple ratio of constants, which is much easier to evaluate. This simplification is a common strategy in limit evaluation, and mastering it is essential for solving a wide range of problems. The ability to identify and handle terms that approach zero as x approaches infinity is a fundamental skill in calculus, and this step provides a clear illustration of its application.

Therefore, the limit becomes:

${ = \frac{0+0}{2-0+0} = \frac{0}{2} = 0 }$

Thus, the limit of the function as x approaches infinity is 0. This result tells us that as x grows larger and larger, the function ${\frac{8 x+2}{2 x^2-5 x+9}}$ gets closer and closer to zero. This means that the x-axis acts as a horizontal asymptote for the function. In other words, the graph of the function will approach the x-axis but never actually touch it as x goes to infinity. This behavior is characteristic of rational functions where the degree of the denominator is greater than the degree of the numerator. The denominator grows faster than the numerator, causing the fraction to shrink towards zero. This understanding is crucial in various applications, such as analyzing the long-term behavior of systems modeled by such functions. For example, in physics, this might represent the decay of a signal over time, or in economics, it could represent the diminishing impact of a policy as time progresses. The fact that the limit is 0 provides valuable information about the function's asymptotic behavior, allowing us to make predictions and draw conclusions about the system it represents. This interpretation is a key aspect of understanding limits and their practical significance.

Conclusion: Significance of Limits at Infinity

In conclusion, evaluating the limit ${\lim _{x \rightarrow \infty} \frac{8 x+2}{2 x^2-5 x+9}}$ demonstrates a fundamental technique in calculus for analyzing the behavior of functions as their input grows infinitely large. By dividing both the numerator and denominator by the highest power of x in the denominator, we effectively simplified the expression, revealing that the limit is 0. This process not only provides the answer but also enhances our understanding of how rational functions behave at extreme values. The result signifies that as x becomes increasingly large, the function approaches zero, indicating a horizontal asymptote at y = 0. This concept is crucial for understanding the end behavior of functions and sketching their graphs accurately. Furthermore, the ability to evaluate limits at infinity has far-reaching applications in various fields, including physics, engineering, and economics. In physics, it can be used to analyze the long-term behavior of physical systems. In engineering, it is essential for designing stable control systems. In economics, it can help in predicting market trends and the long-term effects of economic policies. The skills developed in evaluating limits at infinity are therefore indispensable for anyone pursuing a career in these fields. Mastering these techniques equips us with the tools to analyze complex systems and make informed decisions based on their predicted behavior. This example serves as a cornerstone for understanding more advanced concepts in calculus and beyond, highlighting the power and elegance of mathematical analysis in solving real-world problems. The understanding of limits at infinity is not just an academic exercise but a practical skill that enables us to model and understand the world around us.