Calculating Limits: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of limits, specifically tackling the limit of the function (3x - 1)² - 4 over x² + 4x - 5 as x approaches 1. This might seem a bit intimidating at first, but trust me, we'll break it down step by step. By the end of this guide, you'll not only be able to solve this problem but also have a better understanding of how limits work in general. Ready to get started? Let's go!

Understanding the Problem: The Limit of a Function

So, what exactly are we trying to do? We want to find the limit of the function (3x - 1)² - 4 over x² + 4x - 5 as x gets incredibly close to 1. Think of it like this: We're asking, "What value does this function get closer and closer to as x gets closer and closer to 1?" The key here is that x doesn't actually have to equal 1; it just needs to get infinitely close. This is crucial because, as we'll see, plugging in x = 1 directly might cause some issues (like division by zero). We will explore different methods for calculating limits, which are core concepts in calculus, and understanding them is fundamental for further explorations in math and physics. Limits are not just about getting an answer; they are about understanding the behavior of functions near specific points, which helps us understand continuity, derivatives, and other advanced topics. For instance, the concept of a derivative heavily relies on limits; it essentially defines the slope of a function at a specific point using the limit definition. Similarly, when studying integrals, you will constantly rely on limits, for example, the definite integral is defined as the limit of a sum. Therefore, it is essential that you become comfortable with the idea of limits as they help us predict the behavior of a function. The ability to manipulate and analyze these functions is a critical skill in many STEM fields, including engineering, computer science, and economics. We will look at this function which is a fraction, and in calculus, it is important to find limits of rational functions like this one. This is something that you can definitely do, so pay close attention.

Initial Attempt: Direct Substitution and Its Pitfalls

The first thing you might try is direct substitution. You simply plug in x = 1 into the function and see what you get. So, let's give it a shot:

((3 * 1) - 1)² - 4 over (1)² + 4(1) - 5 which simplifies to (2)² - 4 over 1 + 4 - 5, resulting in 0 over 0. Uh oh! We've got an indeterminate form: 0/0. This tells us that direct substitution won't work, at least not right away. This is where our understanding of limits comes into play. The form 0/0 doesn't mean the limit doesn't exist; it simply means we need to do some more work to find out what's going on. It's a signal that there's more to explore! We will have to try another way. This is an example of a rational function, and often these will have holes where the function is not defined. This is a classic problem in calculus, and it tests your skills in algebra. Being able to recognize these indeterminate forms and apply the correct techniques to resolve them is a critical skill to acquire. Think of direct substitution as the first line of defense. If it works, great! But if it fails (like in this case), you'll need to employ other methods to crack the problem. Also, there is nothing wrong with failing here. This is part of the learning process, and it helps us learn new techniques. Remember that this is not the final answer, but it is what we get from the beginning. We just need to refine it so that it can provide us with a useful and correct solution.

Factoring and Simplifying: The Key to Unlocking the Limit

Since direct substitution failed, we'll turn to a common strategy: factoring. The goal here is to simplify the function so we can get rid of the x - 1 term in the denominator that's causing the 0/0 issue. Let's start by expanding the numerator and factoring both the numerator and the denominator:

Expanding and Simplifying the Numerator

First, let's expand the numerator: (3x - 1)² - 4 = (9x² - 6x + 1) - 4 = 9x² - 6x - 3. Now, we can factor out a 3 from the numerator, so we get 3(3x² - 2x - 1). This makes things a bit cleaner.

Factoring the Denominator

Next, we factor the denominator: x² + 4x - 5. We're looking for two numbers that multiply to -5 and add to 4. Those numbers are 5 and -1. So, the denominator factors into (x + 5)(x - 1).

Putting it All Together

Now, let's rewrite our original function with these factored forms: 3(3x² - 2x - 1) over (x + 5)(x - 1). Wait, we can factor the numerator further. The quadratic 3x² - 2x - 1 factors into (3x + 1)(x - 1). The function then becomes: 3(3x + 1)(x - 1) over (x + 5)(x - 1). And here's the magic moment! We can cancel out the (x - 1) term from both the numerator and the denominator. This is crucial because (x - 1) is the term that becomes 0 when x = 1, causing the initial 0/0 problem. The simplification of this function gets us closer to our desired answer. This process is more than just finding a limit; it is about analyzing the function to fully understand its behavior. Understanding how to approach limits like this is very valuable because it lets you visualize and calculate the function. It is important to remember the factoring patterns as they are important in order to simplify these functions. By simplifying the function, we're essentially removing the "hole" or discontinuity at x = 1, allowing us to find the value the function approaches.

Evaluating the Simplified Function: Finding the Limit

After canceling out the (x - 1) term, we're left with the simplified function: 3(3x + 1) over (x + 5). Now, we can try direct substitution again, plugging in x = 1: 3(3 * 1 + 1) over (1 + 5), which becomes 3(4) over 6, which equals 12/6 or 2. Eureka! We found the limit. As x approaches 1, the function (3x - 1)² - 4 over x² + 4x - 5 gets closer and closer to 2. The removal of the discontinuity allows us to find a concrete value. The simplification process enables us to understand the behavior of a function, which is key to solving it. When you are trying to solve a limit problem, always start by substituting x with the number being approached. If this does not work, you can try to simplify it with factoring, as we did, or with other methods such as L'Hopital's rule. The ability to use these mathematical tools will help you further in your calculus journey, allowing you to analyze many functions and their behavior. Remember that the answer is just one piece of the puzzle. The real value lies in understanding the process and how to apply these techniques to other limit problems.

Conclusion: Mastering the Art of Limits

Congratulations, guys! You've successfully calculated the limit of (3x - 1)² - 4 over x² + 4x - 5 as x approaches 1, and the answer is 2. Here's a quick recap of what we did:

1. Initial Attempt: We tried direct substitution, which resulted in an indeterminate form (0/0). This told us we needed a different approach.
2. Factoring and Simplification: We expanded the numerator, factored both the numerator and denominator, and canceled out the problematic (x - 1) term.
3. Evaluating the Simplified Function: We substituted x = 1 into the simplified function, which gave us the final answer of 2.

Final Thoughts and Tips

• Practice Makes Perfect: The more limit problems you solve, the more comfortable you'll become with these techniques. Try different functions, and don't be afraid to make mistakes – they're part of the learning process.
• Recognize Patterns: Learn to spot common factoring patterns and indeterminate forms. This will speed up your problem-solving.
• Explore Different Methods: While factoring is a powerful tool, there are other techniques for evaluating limits, such as L'Hopital's rule (which you'll likely learn in a calculus course). Knowing multiple methods gives you more options! Understanding these different options will allow you to analyze and resolve a wide range of limit problems. Different kinds of problems may require different methods, which is why it is important to know the options. Don't be afraid to try different methods when you are practicing as it will help you further. This will help you refine your skills, making you a proficient limit calculator.

So, keep practicing, keep exploring, and keep challenging yourself. You've got this!