Identifying Horizontal Asymptotes For F(x) = 3/(5x) A Comprehensive Guide

Hey guys! Today, let's dive into finding the horizontal asymptote of the function f(x) = 3/(5x). This might sound intimidating, but trust me, it’s simpler than it looks. Understanding asymptotes is crucial in calculus and helps in graphing functions accurately. So, let's break it down step-by-step and make sure we understand the concept thoroughly. Horizontal asymptotes, in essence, tell us about the function's behavior as x approaches positive or negative infinity. They are the lines that the graph of the function gets closer and closer to but never quite touches or crosses, at least in the extreme values of x. To identify them, we need to analyze what happens to f(x) as x becomes a very large positive number and a very large negative number. Consider the function f(x) = 3/(5x). As x approaches infinity, the denominator 5x also approaches infinity. This means that we have a constant (3) divided by an increasingly large number. What happens in this scenario? Well, the fraction gets smaller and smaller, approaching zero. Similarly, as x approaches negative infinity, 5x approaches negative infinity. So, we have 3 divided by a very large negative number. Again, the fraction gets smaller and smaller, but this time it approaches zero from the negative side. Mathematically, we can express this as follows:

$$\lim_{x \to \infty} \frac{3}{5x} = 0$$

$$\lim_{x \to -\infty} \frac{3}{5x} = 0$$

These limits tell us that as x goes to infinity or negative infinity, f(x) approaches 0. Therefore, the horizontal asymptote of the function is the line y = 0. This means the graph of f(x) will get closer and closer to the x-axis but will never actually touch it as x moves towards extremely large positive or negative values. This understanding is super important for sketching the graph of the function. You know that the graph will hug the x-axis on both ends, giving you a clear idea of the function’s behavior in those regions. Remember, horizontal asymptotes are not always present. Some functions might have one, some might have two (one for positive infinity and another for negative infinity), and some might have none. It all depends on how the function behaves as x approaches infinity. So, by analyzing the behavior of the function f(x) = 3/(5x), we've nailed down its horizontal asymptote, which is y = 0. This is a foundational concept in understanding rational functions, and mastering it will certainly boost your calculus skills!

Understanding Horizontal Asymptotes: A Deep Dive

To really get a handle on identifying horizontal asymptotes, let's explore the concept a bit more deeply. We know that horizontal asymptotes describe the end behavior of a function. But what exactly does that mean? It means we are looking at what happens to the function's y-values as the x-values become extremely large (positive or negative). This end behavior is crucial in many applications, from physics to economics, where we need to understand long-term trends. In our example, f(x) = 3/(5x), we saw that as x grows without bound, the function approaches zero. This is a classic case of a rational function where the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator. The degree of a polynomial is simply the highest power of x. In our function, the numerator is a constant (3), which can be thought of as 3x^0, so its degree is 0. The denominator is 5x, which has a degree of 1. When the denominator's degree is higher, the function will always approach zero as x goes to infinity or negative infinity. This is because the denominator grows much faster than the numerator, making the overall fraction smaller and smaller. However, the situation changes when the degrees of the numerator and denominator are the same, or when the degree of the numerator is higher. For instance, consider a function like g(x) = (2x^2 + 1) / (x^2 + 3). Here, both the numerator and denominator have a degree of 2. To find the horizontal asymptote in this case, we look at the leading coefficients, which are the coefficients of the highest power of x. In this function, the leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. The horizontal asymptote is the ratio of these coefficients, which is y = 2/1 = 2. So, as x approaches infinity or negative infinity, g(x) approaches 2. If the degree of the numerator is higher than the degree of the denominator, there is no horizontal asymptote. Instead, the function might have a slant (or oblique) asymptote, which is a linear function that the graph approaches. These types of asymptotes are a bit more complex to find, often requiring polynomial division. Understanding these rules helps us quickly identify horizontal asymptotes for various rational functions without needing to calculate limits every time. But it's still important to remember the fundamental idea of looking at the function's behavior as x approaches infinity. This provides a solid foundation for more advanced calculus concepts and applications.

Step-by-Step Solution: Identifying the Horizontal Asymptote

Alright, let's break down the process of identifying the horizontal asymptote of f(x) = 3/(5x) step-by-step. This way, you'll have a clear methodology to apply to similar problems. First, it’s crucial to recognize that we are dealing with a rational function. Rational functions are functions that can be written as the ratio of two polynomials. In our case, the numerator is the constant polynomial 3, and the denominator is the linear polynomial 5x. Now that we've identified the type of function, we can proceed with the steps for finding the horizontal asymptote. Step 1: Analyze the degrees of the polynomials. As we discussed earlier, the degree of a polynomial is the highest power of x. In the numerator, we have a constant, which can be thought of as 3x^0. So, the degree of the numerator is 0. In the denominator, we have 5x, which can be written as 5x^1. So, the degree of the denominator is 1. Step 2: Compare the degrees. We see that the degree of the denominator (1) is greater than the degree of the numerator (0). This is a key observation because it tells us something important about the horizontal asymptote. Step 3: Apply the rule for horizontal asymptotes. When the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is always y = 0. This rule is derived from the fact that as x becomes very large, the denominator grows much faster than the numerator, causing the fraction to approach zero. Step 4: State the horizontal asymptote. Based on our analysis, the horizontal asymptote of f(x) = 3/(5x) is y = 0. This means the graph of the function will get closer and closer to the x-axis as x approaches positive or negative infinity. To solidify your understanding, you can visualize this by graphing the function. You'll notice that the graph hugs the x-axis on both the left and right sides, confirming that y = 0 is indeed the horizontal asymptote. This step-by-step approach can be applied to any rational function. By analyzing the degrees of the polynomials and applying the appropriate rules, you can quickly and accurately identify horizontal asymptotes. It’s all about recognizing the patterns and understanding the underlying principles.

Choosing the Correct Option: A Walkthrough

Now that we've thoroughly analyzed the function f(x) = 3/(5x) and identified its horizontal asymptote as y = 0, let’s walk through how to choose the correct option from the given choices. This is an essential skill, especially when tackling multiple-choice questions in exams. The options provided are:

A. y = 3/5 B. y = 0 C. y = 5/3 D. No horizontal asymptote

We've already determined that the horizontal asymptote is y = 0, so we are looking for the option that matches this. Let's go through each option systematically:

• Option A: y = 3/5 This option represents a horizontal line at y = 3/5. We know from our analysis that the function approaches 0 as x approaches infinity, so this option is incorrect.
• Option B: y = 0 This is exactly what we found as the horizontal asymptote. As x approaches infinity, f(x) = 3/(5x) approaches 0. So, this option is correct.
• Option C: y = 5/3 This option represents another horizontal line, but at y = 5/3. This is the reciprocal of the coefficient we see in the function, but it doesn't represent the end behavior of the function. So, this option is incorrect.
• Option D: No horizontal asymptote* This option is incorrect because we have clearly shown that the function does have a horizontal asymptote, and it is y = 0.

Therefore, by carefully considering each option and comparing it to our analysis, we can confidently choose the correct answer. The correct option is B. y = 0. This exercise highlights the importance of not just finding the answer but also understanding why the other options are incorrect. This deeper understanding strengthens your grasp of the concept and helps you avoid common mistakes. When solving similar problems, always take the time to consider all the options and eliminate the ones that don't fit the criteria. This strategy will significantly improve your accuracy and confidence in answering questions about horizontal asymptotes and other mathematical concepts.

Practice Problems: Test Your Understanding

To truly master the concept of horizontal asymptotes, it's essential to practice with a variety of problems. Practice helps solidify your understanding and builds confidence in your ability to apply the rules. So, let’s look at a few practice problems to test your skills. Working through these will give you a better grasp of how to identify horizontal asymptotes in different scenarios. Problem 1: Find the horizontal asymptote of the function g(x) = (4x + 1) / (2x - 3). This function is a rational function, just like our example, but this time, the degrees of the numerator and denominator are the same. Remember the rule for this situation: when the degrees are equal, you need to look at the leading coefficients. The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 2. So, the horizontal asymptote will be the ratio of these coefficients. Can you figure it out? Problem 2: Identify the horizontal asymptote of the function h(x) = (x^2 + 2x + 1) / (x^3 - 4x). In this case, the degree of the denominator is greater than the degree of the numerator. We know what happens in this scenario, right? The function will approach zero as x goes to infinity or negative infinity. So, what’s the horizontal asymptote? Problem 3: Determine the horizontal asymptote of the function k(x) = (3x^2 - 5) / (x^2 + 2x). Again, we have a rational function, but this time, the degrees of the numerator and denominator are the same. This means we need to focus on the leading coefficients. What are they, and what’s their ratio? Problem 4: What is the horizontal asymptote of the function p(x) = (2x^3 + x) / (x^2 - 1)? This function is a bit different because the degree of the numerator is greater than the degree of the denominator. What does this tell us about the horizontal asymptote? Does it exist, or is there something else we should be looking for? Try working through these problems on your own. Remember to follow the steps we discussed: identify the type of function, analyze the degrees of the polynomials, apply the appropriate rules, and state the horizontal asymptote. If you get stuck, go back and review the explanations and examples we've covered. The solutions to these practice problems are:

• Problem 1: y = 4/2 = 2
• Problem 2: y = 0
• Problem 3: y = 3/1 = 3
• Problem 4: No horizontal asymptote (there is a slant asymptote)

By tackling these problems, you'll reinforce your understanding of horizontal asymptotes and be well-prepared to handle more complex functions. Keep practicing, and you'll become a pro at identifying asymptotes in no time!