Solving Limits Step-by-Step Limit X Approaches 3 (X²-9)/(√X²+7-4)

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Hey guys! Today, we're going to break down a classic calculus problem: solving the limit of (X²-9)/(√X²+7-4) as X approaches 3. This might look intimidating at first, but don't worry, we'll go through it step by step, making sure you understand every little detail. Limits are a fundamental concept in calculus, so mastering them is crucial for your mathematical journey. Let's get started!

Understanding the Problem

Before we dive into the solution, let's first understand what a limit actually means. In simple terms, a limit tells us what value a function approaches as the input (in this case, X) gets closer and closer to a specific value (in this case, 3). It's not necessarily the value of the function at that point, but rather where the function is heading. When we encounter a limit problem like this, the first thing we usually try is direct substitution. This means plugging in the value that X is approaching (which is 3) directly into the expression. Let's see what happens when we do that:

(3² - 9) / (√(3² + 7) - 4) = (9 - 9) / (√(9 + 7) - 4) = 0 / (√16 - 4) = 0 / (4 - 4) = 0 / 0

Uh oh! We've got 0/0, which is an indeterminate form. This means that direct substitution didn't give us a clear answer. It doesn't mean the limit doesn't exist; it just means we need to use another method to find it. Indeterminate forms are common in limit problems, and they signal that we need to do some algebraic manipulation to simplify the expression before we can evaluate the limit. The most common techniques include factoring, rationalizing the numerator or denominator, or using L'Hôpital's Rule (which we won't need for this particular problem, but it's good to keep in mind for other cases). For this problem, we'll use a clever trick: rationalizing the denominator. This involves multiplying both the numerator and denominator by the conjugate of the denominator, which will help us get rid of the square root and simplify the expression.

The Strategy: Rationalizing the Denominator

The main reason why we got the indeterminate form is because of the square root in the denominator. To get rid of it, we'll use a technique called "rationalizing the denominator." This involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of √X²+7-4 is √X²+7+4. Remember, multiplying by the conjugate is like multiplying by 1, because we're essentially multiplying by (√X²+7+4)/(√X²+7+4), which doesn't change the value of the expression. Here’s how it works:

lim X->3 (X²-9)/(√X²+7-4) * (√X²+7+4)/(√X²+7+4)

Now, we need to carefully multiply both the numerator and the denominator. In the numerator, we'll just leave the expression as a product for now. In the denominator, we'll use the difference of squares formula: (a - b)(a + b) = a² - b². This will eliminate the square root. Let's see how it plays out:

Numerator: (X²-9)(√X²+7+4)

Denominator: (√X²+7-4)(√X²+7+4) = (√X²+7)² - 4² = (X² + 7) - 16 = X² - 9

Notice something cool? We now have (X² - 9) in both the numerator and the denominator! This means we can cancel them out, which will simplify the expression even further. This is exactly what we were aiming for when we rationalized the denominator. By getting rid of the square root and simplifying the expression, we're one step closer to evaluating the limit. This is a common strategy in calculus: to manipulate the expression algebraically until we can get rid of the indeterminate form and directly substitute the value.

Step-by-Step Solution

Okay, let's walk through the solution step-by-step, so you can see exactly how it all comes together:

  1. Start with the original limit:

lim X->3 (X²-9)/(√X²+7-4)

  1. Multiply the numerator and denominator by the conjugate of the denominator:

lim X->3 (X²-9)/(√X²+7-4) * (√X²+7+4)/(√X²+7+4)

  1. Expand the denominator using the difference of squares:

lim X->3 (X²-9)(√X²+7+4) / ((√X²+7)² - 4²)

  1. Simplify the denominator:

lim X->3 (X²-9)(√X²+7+4) / (X² + 7 - 16)

lim X->3 (X²-9)(√X²+7+4) / (X² - 9)

  1. Cancel out the (X² - 9) terms:

lim X->3 (√X²+7+4)

  1. Now we can directly substitute X = 3:

√3²+7+4 = √9+7+4 = √16+4 = 4+4 = 8

So, the limit of (X²-9)/(√X²+7-4) as X approaches 3 is 8. Awesome, right? We took a problem that seemed tricky at first and, by using a clever algebraic technique, we simplified it and found the answer. Remember, the key was to identify the indeterminate form and then use a method to get rid of it. In this case, rationalizing the denominator was the perfect tool for the job. But don't stop here! The more you practice these types of problems, the more comfortable you'll become with different techniques and the better you'll get at recognizing which method to use in each situation.

Factoring Approach

Another way we could tackle this problem, though not as direct in this case, is by factoring the numerator. Remember that X² - 9 is a difference of squares, which can be factored as (X - 3)(X + 3). While this doesn't immediately solve the problem because we still have the square root in the denominator, it's a good habit to look for factoring opportunities, as it often leads to simplification. If we rewrite the limit with the factored numerator, we get:

lim X->3 ((X - 3)(X + 3))/(√X²+7-4)

Now, we still have the issue of the indeterminate form 0/0 when we substitute X = 3. Factoring the numerator alone didn't solve the problem this time, but it's an important technique to keep in your toolbox. In some cases, factoring can directly lead to cancellation and simplification. In this specific problem, we still need to rationalize the denominator as the next step, but recognizing the possibility of factoring is a valuable skill.

Key Takeaways and Practice

So, what are the key takeaways from this problem? First, always try direct substitution first. If you get an indeterminate form like 0/0, it means you need to do more work. Second, rationalizing the denominator (or numerator) is a powerful technique when dealing with square roots in limits. Third, don't forget about factoring! It might not always be the first step, but it's a valuable tool for simplifying expressions. And finally, practice, practice, practice! The more you work through limit problems, the better you'll become at recognizing patterns and applying the right techniques.

Limits are a cornerstone of calculus, and mastering them will set you up for success in more advanced topics. So, keep practicing, keep exploring different techniques, and don't be afraid to ask questions. Math can be challenging, but it's also incredibly rewarding when you finally solve a tough problem. You got this!

To solidify your understanding, try working through similar problems. Look for examples online or in your textbook. Try changing the value that X approaches or modifying the function itself. The more you experiment, the better you'll understand the concepts. And remember, if you get stuck, don't hesitate to seek help from your teacher, classmates, or online resources. Learning math is a journey, and we're all in this together.

Keywords: Limit problems, solving limits, step-by-step solutions, indeterminate form, rationalizing denominator