Limit Of (x² + 6x + 9)/(x² + 2x - 3) As X Approaches Infinity

Hey guys! Let's dive into a fun math problem today. We're going to figure out the limit of the function (x² + 6x + 9)/(x² + 2x - 3) as x gets super huge – we're talking infinity here! And don't worry, we'll break it down step by step so it's super easy to understand. We've got some multiple-choice options: A) 1, B) 2, C) 3, and D) Infinity. By the end of this, you'll not only know the answer but also why it's the answer. Let's get started!

Understanding Limits and Infinity

Before we jump right into solving this specific problem, let's make sure we're all on the same page about what limits and infinity actually mean in math. It's like building the foundation for a strong understanding, you know? So, what exactly is a limit? Basically, when we talk about the limit of a function as x approaches a certain value (in our case, infinity), we're asking: What value does the function get closer and closer to as x gets closer and closer to that value? It's not necessarily about what the function is at that exact point, but more about its behavior as it approaches that point.

Now, let's tackle infinity. Infinity isn't a number, guys. It's more like a concept. It represents something that goes on forever and ever without any end. When we say x approaches infinity, we mean x is getting bigger and bigger without any bound. Think of it like a never-ending road stretching out into the distance. So, when we're trying to find the limit as x approaches infinity, we're trying to figure out what happens to our function as x becomes incredibly, unbelievably large. What value does the function start to cozy up to? This understanding is crucial because it helps us simplify complex expressions and focus on what really matters as x gets huge. Keep this in mind as we move forward, and you'll see how this concept makes solving the problem way easier!

Identifying Dominant Terms

Alright, so we've got the basics of limits and infinity down. Now, let's talk about something super important for solving our problem: dominant terms. These are the VIPs of our function when x gets super big. Think of it like this: imagine you're adding a tiny pebble to a giant mountain of sand. Does the pebble really change the size of the mountain much? Nope! It's the same idea with dominant terms.

In a polynomial expression (like the ones we have in our function), the dominant term is the one with the highest power of x. Why? Because as x gets incredibly large, that term will grow much faster than any other term. The other terms become relatively insignificant, like that pebble on the mountain. Let's look at our function again: (x² + 6x + 9)/(x² + 2x - 3). In the numerator (the top part), we have x² + 6x + 9. Which term has the highest power of x? It's x², right? So, x² is the dominant term in the numerator. Similarly, in the denominator (the bottom part), x² + 2x - 3, the dominant term is also x². Now, why is this so helpful? Because when x approaches infinity, we can focus on these dominant terms and pretty much ignore the rest! They're the ones calling the shots and determining the function's behavior. This makes our lives much easier because we can simplify the problem and see the big picture more clearly. Trust me, identifying dominant terms is a game-changer for limit problems!

Simplifying the Function

Okay, we've identified the dominant terms, which is awesome! Now comes the fun part: simplifying our function using this knowledge. Remember, the original function is (x² + 6x + 9)/(x² + 2x - 3). We figured out that x² is the dominant term in both the numerator and the denominator. So, as x approaches infinity, we can essentially focus on these terms and kind of forget about the others. It's like saying, "Hey, 6x and 9, you guys are cool, but x² is the star of the show right now!" and the same goes for the denominator.

So, we can approximate our function as x² / x² when x is incredibly large. See how much simpler that is? We've gone from a somewhat complicated expression to a super straightforward one. Now, what happens when you divide something by itself? It equals 1, right? So, x² / x² simplifies to 1. This is a huge step! We've taken our original function and, by focusing on the dominant terms, we've simplified it to a constant value. This tells us something really important: as x gets bigger and bigger, the function gets closer and closer to 1. It's like the function is aiming for 1 as its final destination. This simplification is the key to solving the limit problem, and it all comes down to understanding those dominant terms. Now, let's see what this means for our answer choices!

Determining the Limit

Alright, we've done the heavy lifting! We simplified the function (x² + 6x + 9)/(x² + 2x - 3) by focusing on the dominant terms, and we found that as x approaches infinity, the function behaves like x² / x², which simplifies to 1. Now comes the moment of truth: what does this tell us about the limit? Well, the limit of a function as x approaches infinity is the value that the function gets closer and closer to as x gets larger and larger. And we just showed that our function gets closer and closer to 1. So, drumroll please... the limit of the function as x approaches infinity is 1!

Let's look back at our multiple-choice options: A) 1, B) 2, C) 3, and D) Infinity. Which one matches our result? Option A) 1, of course! We've nailed it! But it's not just about getting the right answer, it's about understanding why it's the right answer. We saw how identifying the dominant terms allowed us to simplify the problem and make it much more manageable. We also reinforced the idea that a limit describes the function's behavior as it approaches a value, not necessarily the value it is at that point. So, not only do you now know the answer to this specific problem, but you also have a powerful tool for tackling similar limit problems in the future. Go you!

Final Answer

Okay, guys, let's wrap this up and make sure we're crystal clear on everything. We started with the question: What is the limit of the function (x² + 6x + 9)/(x² + 2x - 3) as x approaches infinity? We had some options: A) 1, B) 2, C) 3, and D) Infinity. We've gone through the whole process step by step, and we've arrived at our final answer.

By identifying the dominant terms (x² in both the numerator and denominator), we simplified the function to x² / x². This then simplified further to 1. This means that as x gets incredibly large, the function gets closer and closer to the value 1. Therefore, the limit of the function as x approaches infinity is 1.

So, the final answer is A) 1. We didn't just guess; we used our understanding of limits, infinity, and dominant terms to solve the problem logically and confidently. You now have a solid understanding of how to approach these types of problems, and that's what really matters. You've got this! Now go out there and conquer some more math challenges!