Solving Limits A Step-by-Step Guide To Lim X→2 (x^2-4)/(x^2-5x+6)

by ADMIN 66 views

Hey guys! Ever find yourself staring at a math problem that seems like it's written in another language? Limits in calculus can feel that way at first glance, but trust me, they're super cool once you get the hang of them. Today, we're going to break down a classic limit problem step by step. We're talking about finding the value of the limit as x approaches 2 for the function (x^2 - 4) / (x^2 - 5x + 6). Sounds intimidating, right? Don't sweat it! We'll make it crystal clear, so you'll be solving these like a pro in no time. Calculus often deals with functions and their behavior, and one crucial concept in calculus is that of a limit. Limits help us understand what happens to a function as its input approaches a certain value. They form the foundation of calculus, underpinning ideas like continuity, derivatives, and integrals. They allow us to analyze the behavior of functions, especially near points where the function might not be directly defined. Understanding limits is essential for tackling more complex calculus problems and real-world applications. Limits help us analyze rates of change, optimize functions, and model physical phenomena. So, let's dive into the fascinating world of limits and conquer this problem together!

Before we jump into the problem, let's quickly recap what limits are all about. In simple terms, a limit tells us what value a function approaches as its input gets closer and closer to a specific point. Imagine you're walking towards a door. The limit is like figuring out where you'll end up as you get right up to the doorway. It's not necessarily about where you are at the door, but where you're heading. The concept of limits can be a bit tricky at first. We're not just plugging in a value and getting an answer. Instead, we're looking at the trend—the direction the function is heading. Think of it like predicting the weather. You're not saying exactly what will happen, but you're making an educated guess based on current conditions and trends. In calculus, we often use limits to handle situations where direct substitution doesn't work. For example, if we try to plug in a value that makes the denominator of a fraction zero, we end up with an undefined expression. Limits help us get around this by allowing us to examine the function's behavior as we get infinitesimally close to that problematic value. So, when you hear "limit," think about approaching, trending, and predicting. It's about understanding the behavior of functions, especially when things get a little tricky. Now, with this basic understanding in place, let's tackle our specific problem and see how limits work in action. We'll break it down step by step, so you'll see exactly how to solve it.

Okay, let's get down to business. Our mission is to find the value of the limit as x approaches 2 for the function (x^2 - 4) / (x^2 - 5x + 6). This looks like a mouthful, but we'll take it one step at a time. The problem is stated as: lim x→2 (x^2 - 4) / (x^2 - 5x + 6). This notation tells us we want to find the limit of the function as x gets closer and closer to 2. Remember, we're not plugging in 2 just yet. We're exploring what the function does as x gets really, really close to 2. First things first, let's try the most straightforward approach: direct substitution. What happens if we just plug in x = 2 into the function? If we substitute x = 2 directly into the expression, we get: (2^2 - 4) / (2^2 - 5*2 + 6). Evaluating this gives us (4 - 4) / (4 - 10 + 6), which simplifies to 0 / 0. Uh-oh! We've hit a snag. 0 / 0 is an indeterminate form. This means we can't determine the limit just by plugging in the value. It's like hitting a roadblock on our journey to find the limit. So, direct substitution didn't work this time. But don't worry, this is a common situation in limit problems, and it just means we need to use a different strategy. The fact that we got an indeterminate form tells us there might be some simplification or manipulation we can do to the function. So, what's our next move? We need to find a way to rewrite the expression so we can evaluate the limit. Keep your thinking caps on; we're about to dive into some algebra!

Since direct substitution gave us an indeterminate form, we need to try a different approach. Factoring is often the key to simplifying expressions like this. Let's start by factoring the numerator and the denominator separately. The numerator is x^2 - 4. This is a classic difference of squares, which factors into (x - 2)(x + 2). Remember the formula: a^2 - b^2 = (a - b)(a + b). In our case, a = x and b = 2, so it fits perfectly. Now, let's tackle the denominator: x^2 - 5x + 6. We need to find two numbers that multiply to 6 and add up to -5. Those numbers are -2 and -3. So, the denominator factors into (x - 2)(x - 3). Factoring might seem like a small step, but it's a crucial one. It often reveals hidden structures and common factors that can simplify the expression. Think of it like organizing a messy room. Once everything is sorted and in its place, you can see the underlying order. So, our original expression (x^2 - 4) / (x^2 - 5x + 6) can now be rewritten as [(x - 2)(x + 2)] / [(x - 2)(x - 3)]. This is progress! We've transformed the expression into a form that might be easier to work with. Notice anything interesting? We've got a common factor in both the numerator and the denominator. This is a sign that we're on the right track. Factoring helps us identify these common factors, which we can then cancel out to simplify the expression further. So, let's move on to the next step and see how this factoring helps us solve the limit problem.

Alright, we've factored the numerator and denominator, and now we have the expression [(x - 2)(x + 2)] / [(x - 2)(x - 3)]. Take a good look at it. Do you see any common factors that we can cancel out? You got it! We have (x - 2) in both the numerator and the denominator. This means we can simplify the expression by dividing both the top and the bottom by (x - 2). Remember, we're working with limits, which means we're looking at what happens as x approaches 2, but not exactly at x = 2. So, as long as x isn't actually equal to 2, we can safely cancel out the (x - 2) terms. Once we cancel out the common factor, our expression simplifies to (x + 2) / (x - 3). This is a much simpler form than what we started with! It's like taking a complex puzzle and reducing it to its essential pieces. Now, the expression looks much more manageable, and we can see a clearer path to finding the limit. This step of simplifying is crucial in many limit problems. It helps us eliminate indeterminate forms and makes the expression easier to evaluate. By canceling out common factors, we're essentially removing the source of the problem that was causing the 0 / 0 situation. So, where do we go from here? We've simplified the expression; now it's time to see if we can evaluate the limit using our new, simplified form. Let's move on to the next step and find out!

Okay, we've done the hard work of factoring and simplifying, and now we're at the final stretch. Our simplified expression is (x + 2) / (x - 3). Remember, we're trying to find the limit as x approaches 2. So, let's try direct substitution again, but this time with our simplified expression. We substitute x = 2 into (x + 2) / (x - 3), which gives us (2 + 2) / (2 - 3). Evaluating this, we get 4 / (-1), which simplifies to -4. We have a result! Unlike our first attempt, we didn't end up with an indeterminate form. We got a definite value: -4. This means that as x gets closer and closer to 2, the value of the function (x^2 - 4) / (x^2 - 5x + 6) approaches -4. Think of it like this: if you were to graph the function, you'd see that as you move along the graph towards the point where x = 2, the y-value gets closer and closer to -4. So, we've successfully found the limit! The limit as x approaches 2 of (x^2 - 4) / (x^2 - 5x + 6) is -4. This is a great example of how factoring and simplifying can help us evaluate limits that initially seem impossible to solve. Direct substitution is always the first thing to try, but when it leads to an indeterminate form, we need to use other techniques, like factoring, to rewrite the expression. And that's exactly what we did here. Now, let's summarize our steps and make sure we've got the whole process down.

Let's quickly recap the steps we took to solve this limit problem. This will help solidify your understanding and give you a clear roadmap for tackling similar problems in the future. Here's the breakdown:

  1. State the Problem: We started with the limit problem: lim x→2 (x^2 - 4) / (x^2 - 5x + 6).
  2. Try Direct Substitution: We first attempted to plug in x = 2 directly into the function. This gave us 0 / 0, an indeterminate form, which meant we needed a different approach.
  3. Factor the Numerator and Denominator: We factored the numerator (x^2 - 4) into (x - 2)(x + 2) and the denominator (x^2 - 5x + 6) into (x - 2)(x - 3). Factoring is a powerful technique for simplifying expressions and revealing hidden structures.
  4. Simplify the Expression: We canceled out the common factor (x - 2) from the numerator and denominator, which simplified the expression to (x + 2) / (x - 3). Simplifying is key to making the expression easier to evaluate.
  5. Evaluate the Limit: We substituted x = 2 into the simplified expression (x + 2) / (x - 3), which gave us (2 + 2) / (2 - 3) = 4 / (-1) = -4. This gave us our final answer.

So, the value of the limit is -4. By following these steps, we were able to break down a seemingly complex problem into manageable parts and arrive at the solution. Remember, the key to solving limit problems is to try direct substitution first, and if that doesn't work, look for ways to factor and simplify the expression. With practice, you'll become a pro at solving these types of problems!

Alright guys, we've reached the end of our journey through this limit problem. We started with a seemingly daunting expression and, step by step, broke it down and found the solution. We've shown that the limit of (x^2 - 4) / (x^2 - 5x + 6) as x approaches 2 is -4. This problem illustrates some important principles in calculus. Direct substitution is our first port of call, but when it leads to an indeterminate form, we need to roll up our sleeves and use our algebraic toolkit. Factoring, simplifying, and then re-evaluating are common strategies for tackling limit problems. The concept of limits is foundational in calculus, and mastering these techniques is crucial for understanding more advanced topics. Limits are the building blocks for derivatives, integrals, and a whole host of other concepts that are used in everything from physics and engineering to economics and computer science. So, understanding limits isn't just about solving math problems; it's about gaining a deeper insight into how the world works. Keep practicing, keep exploring, and don't be afraid to tackle challenging problems. Each problem you solve is a step forward in your mathematical journey. And remember, limits might seem tricky at first, but with the right approach, they're totally conquerable. So, go forth and conquer those limits! You've got this!