Evaluate Limit As X Approaches Infinity Of (3x^2 + X + 2) / (x^2 + 1)
Introduction to Limits at Infinity
In the realm of calculus, understanding the behavior of functions as their input approaches infinity is a crucial concept. Limits at infinity help us analyze the end behavior of functions, providing insights into how a function behaves for extremely large values of its input. This analysis is not merely an academic exercise; it has practical applications in various fields, including physics, engineering, and economics, where we often need to model and predict the behavior of systems under extreme conditions. In this comprehensive guide, we will delve into the evaluation of limits at infinity, focusing specifically on rational functions. Our primary example will be the function , which we will analyze as x approaches infinity. This exploration will not only provide a step-by-step solution to this particular limit but also equip you with the tools and understanding necessary to tackle similar problems. The concept of a limit is foundational to calculus, serving as the basis for derivatives, integrals, and other advanced topics. When we talk about the limit of a function as x approaches infinity, we are essentially asking: what value does the function approach as x becomes arbitrarily large? To answer this question effectively, we need to understand how different components of the function behave as x grows without bound. For polynomial functions, the term with the highest power of x typically dominates the behavior of the function as x approaches infinity. This is because higher powers of x grow much faster than lower powers. For rational functions, which are ratios of polynomials, the situation is slightly more complex. The limit at infinity depends on the relationship between the degrees of the polynomials in the numerator and the denominator. If the degrees are the same, as in our example, the limit is determined by the ratio of the leading coefficients. If the degree of the numerator is less than the degree of the denominator, the limit is zero. And if the degree of the numerator is greater than the degree of the denominator, the limit is either infinity or negative infinity, depending on the signs of the leading coefficients and the parity of the exponents. Understanding these rules and being able to apply them effectively is key to mastering limits at infinity. In the following sections, we will break down the process of evaluating the limit of as x approaches infinity, providing detailed explanations and justifications for each step. By the end of this guide, you will have a solid understanding of how to approach these types of problems and a deeper appreciation for the power and elegance of calculus.
Problem Statement: Evaluating the Limit
Our primary goal is to evaluate the limit of the function as x approaches infinity. This problem is a classic example of limit evaluation in calculus, and it provides a valuable opportunity to apply key concepts and techniques. To begin, let's restate the problem formally: we want to find
This notation represents the value that the function approaches as x becomes infinitely large. At first glance, it might not be immediately obvious what this limit is. Both the numerator and the denominator grow without bound as x approaches infinity, so we have an indeterminate form of the type . This means we cannot simply substitute infinity into the expression and obtain a meaningful answer. Instead, we need to employ a more sophisticated technique to evaluate the limit. The standard approach for dealing with limits of rational functions at infinity is to divide both the numerator and the denominator by the highest power of x that appears in the denominator. In this case, the highest power of x in the denominator is x2. This technique allows us to simplify the expression and rewrite it in a form where the limit can be more easily determined. By dividing both the numerator and the denominator by x2, we effectively scale the function and examine its behavior as x becomes very large. This process helps to reveal the dominant terms in the numerator and the denominator, which ultimately determine the limit. It's important to note that this technique is valid because dividing both the numerator and the denominator by the same quantity does not change the value of the fraction. We are simply rewriting the function in an equivalent form that is more amenable to limit evaluation. In the next section, we will perform this division and simplify the expression, paving the way for a straightforward calculation of the limit. This step-by-step approach ensures that we not only arrive at the correct answer but also gain a deeper understanding of the underlying principles of limit evaluation.
Step-by-Step Solution
To evaluate the limit, we'll follow the standard procedure for rational functions at infinity. This involves dividing both the numerator and the denominator by the highest power of x present in the denominator. In our case, that's x2. Let's break down the steps:
- Divide by the Highest Power of x: Divide both the numerator and the denominator of the function by x2:
- Simplify the Expression: Now, we simplify the fractions in both the numerator and the denominator:
This step is crucial because it transforms the original function into a form where the limit can be easily evaluated. The terms and in the numerator, and in the denominator, will all approach zero as x approaches infinity. This is a key property of limits that we will exploit in the next step.
- Apply the Limit: Now we can apply the limit as x approaches infinity:
As x approaches infinity, the terms , , and all approach zero. This is because dividing a constant by an increasingly large number results in a value that gets closer and closer to zero. Mathematically, we can write:
- Evaluate the Limit: Substitute these limits into our expression:
Therefore, the limit of the function as x approaches infinity is 3.
This step-by-step approach not only provides the solution but also illustrates the underlying principles of limit evaluation. By dividing by the highest power of x, we effectively isolated the dominant terms in the numerator and the denominator, allowing us to determine the limit with ease. In the next section, we will discuss the result and its implications.
Discussion and Conclusion
Having evaluated the limit, we found that
This result tells us that as x becomes infinitely large, the function approaches the value 3. In other words, the end behavior of the function is to converge towards the horizontal line y = 3. This horizontal line is called a horizontal asymptote of the function. The existence of a horizontal asymptote is a common feature of rational functions, and it provides valuable information about the function's long-term behavior. In this case, the fact that the limit is 3 indicates that the function will get arbitrarily close to 3 as x gets very large, either in the positive or negative direction. The reason for this behavior lies in the fact that the highest power of x in both the numerator and the denominator is x2. As x becomes very large, the terms with lower powers of x (such as x and the constant terms) become insignificant compared to the x2 terms. Therefore, the function behaves approximately like , which simplifies to 3. This intuition can be formalized by the technique we used in the solution, where we divided both the numerator and the denominator by x2. This process effectively isolates the dominant terms and allows us to determine the limit. More generally, for rational functions where the degrees of the numerator and the denominator are the same, the limit at infinity is equal to the ratio of the leading coefficients. In our example, the leading coefficient in the numerator is 3, and the leading coefficient in the denominator is 1, so the limit is . This rule provides a quick way to determine the limit at infinity for many rational functions. In conclusion, evaluating limits at infinity is a fundamental skill in calculus, and it provides valuable insights into the behavior of functions. By understanding the techniques and principles involved, we can analyze the long-term behavior of functions and make predictions about their values under extreme conditions. The example we have explored, , serves as a classic illustration of how to approach these types of problems, and the lessons learned here can be applied to a wide range of similar situations. As we continue our study of calculus, the concept of limits will continue to play a central role, and a solid understanding of limits at infinity will be essential for tackling more advanced topics.