Analyzing Function Asymptotes: A Deep Dive Into F(x) = X + 1/x²

Hey guys! Let's dive into a fascinating math problem that involves analyzing the asymptotes of a function. We'll be focusing on the function f(x) = x + 1/x², which is defined for all real numbers except for zero (denoted as R*). The question asks us to identify the correct statement about its asymptotes from the given options. This is a classic calculus problem that tests your understanding of limits, asymptotes, and function behavior. Let's break it down, shall we?

Understanding the Function and the Question

First off, let's get a clear picture of what we're dealing with. The function f(x) = x + 1/x² is a combination of two parts: a linear term, x, and a rational term, 1/x². The domain of this function excludes zero because division by zero is undefined. This exclusion is a major clue when thinking about asymptotes. The question wants us to pinpoint the number and types of asymptotes this function possesses. Remember, asymptotes are lines that the function approaches but never quite touches as x approaches certain values (or infinity). There are generally three kinds of asymptotes: vertical, horizontal, and oblique (or slant). Let's go over the definitions of each asymptote type to better understand the question.

• Vertical Asymptotes: These occur where the function approaches infinity (positive or negative) as x approaches a specific value. Typically, you'll find these at values where the denominator of a fraction becomes zero. This is a very key thing to think about when analyzing your question and problem.
• Horizontal Asymptotes: These happen when the function approaches a constant value as x approaches positive or negative infinity. This is like the function flattening out at a certain y-value as x gets extremely large or small. Finding this requires a very high understanding of limits.
• Oblique (Slant) Asymptotes: These occur when the function approaches a non-horizontal line as x approaches positive or negative infinity. They arise when the degree of the numerator is exactly one more than the degree of the denominator in a rational function. You can find these by using polynomial division.

So, armed with these definitions, let's get into the specifics of f(x) = x + 1/x².

Analyzing for Vertical Asymptotes

Right, let's start with vertical asymptotes. Vertical asymptotes are usually easy to spot because they pop up where the function is undefined, which is often when you have division by zero. Looking at our function, f(x) = x + 1/x², the denominator x² becomes zero when x = 0. Since our function is undefined at x = 0, this is a good sign for a vertical asymptote. To confirm, we need to examine what happens to f(x) as x approaches 0 from both sides. As x approaches 0 from either side (whether positive or negative), the term 1/x² heads towards positive infinity. The x term, meanwhile, approaches 0. Therefore, the function f(x) approaches positive infinity as x approaches 0. This confirms we have a vertical asymptote at x = 0. Now this narrows down our choices a bit! It indicates that options A and B (which claim no or one asymptote) are incorrect. We can now consider our question more thoughtfully.

Checking for Horizontal Asymptotes

Okay, now let's explore horizontal asymptotes. We need to consider what happens to f(x) as x goes to positive and negative infinity. As x becomes very large (positive or negative), the term 1/x² approaches zero. This is because the denominator x² grows much faster than the numerator 1. So, f(x) = x + 1/x² essentially behaves like f(x) = x as x approaches infinity. This means that f(x) doesn't flatten out to a constant value; it keeps growing (or decreasing). Therefore, we do not have a horizontal asymptote. This piece of information assists us in our quest for an answer.

Investigating Oblique Asymptotes

Alright, let's check for oblique asymptotes. Remember, oblique asymptotes occur when the function behaves like a non-horizontal line as x goes to infinity. We already know that as x approaches infinity, the term 1/x² becomes negligible. Therefore, f(x) behaves approximately like x. The function f(x) = x is a straight line, and as x approaches either positive or negative infinity, f(x) also goes toward positive or negative infinity, matching the behavior of a line. So, the line y = x acts as an oblique asymptote for our function. This is a key insight. Let's recap what we've discovered so far: a vertical asymptote at x = 0 and an oblique asymptote along the line y = x. This now significantly guides our answer.

Deciding on the Correct Answer

So, after all the calculations, we've got the lowdown on the asymptotes. We've got one vertical asymptote at x = 0, and we've determined that y = x is an oblique asymptote. Let's look back at our options:

• A) No asymptotes
• B) One asymptote
• C) Possesses two vertical asymptotes
• D) One oblique asymptote
• E) Possesses two asymptotes

We know there's one vertical asymptote (not two, eliminating C), and we've established the existence of an oblique asymptote (eliminating A and B). So, if we look back at the information we have, the correct answer must be D, which states that it has one oblique asymptote. The correct choice is therefore D. Boom!

Conclusion and Key Takeaways

Alright, folks, we've successfully navigated the analysis of the function f(x) = x + 1/x². We've found a vertical asymptote at x = 0 and an oblique asymptote along the line y = x. Remember that when you approach these types of problems, consider:

• Vertical Asymptotes: Check for values that make the denominator zero.
• Horizontal Asymptotes: Evaluate the function's behavior as x approaches infinity.
• Oblique Asymptotes: See if the function approaches a non-horizontal line.

That's all for now. Keep practicing, and you'll become an asymptote ace in no time! Keep studying and keep learning, guys! Until next time!