Solving Limits: Finding 'k' For Infinite Values

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Hey math enthusiasts! Let's dive into a classic limit problem. We're going to break down how to solve this, step-by-step, making sure even those slightly rusty on their calculus can follow along. Our goal is to find the smallest value of 'k' that satisfies the given limit equation. So, buckle up; it's going to be a fun ride!

Understanding the Problem: The Core of the Limit

Okay, guys, let's get down to the nitty-gritty. The core of this problem revolves around limits, specifically, what happens when 'x' approaches infinity. We're given the following equation: lim x→∞: ((x-1)(x-2)(x-3)(3x-4)) / (kx-5)^4 = (16/3)^-1. Now, what does this even mean? Well, we are trying to figure out what happens to the function as 'x' gets insanely huge – like, bigger than any number you can imagine. The key here is to manipulate this equation to identify the behavior of the highest-degree terms. The function is a ratio of polynomials, and as 'x' tends to infinity, the terms with the highest degree become the most important in determining the limit's value. We need to simplify it, focusing on those all-important leading terms. It's like looking at a race; the frontrunners (highest degree terms) determine the outcome. Getting a handle on how to approach these kinds of problems is essential for anyone who wants to become a pro at calculus. Remember, the limit is the value that the function approaches as the input approaches some value – in this case, infinity. This means that we're not actually evaluating the function at infinity (because that's impossible!), but rather, we're figuring out what value the function gets closer and closer to as 'x' gets larger and larger. Keep in mind that as x approaches infinity, lower-degree terms become negligible in comparison to the highest-degree terms. This is a crucial concept to master when dealing with limits at infinity.

Now, let's break down the given equation and figure out how to solve it. Firstly, the right side of the equation is (16/3)^-1. We can simplify this to 3/16, which is a much friendlier number to work with. Our new equation becomes lim x→∞: ((x-1)(x-2)(x-3)(3x-4)) / (kx-5)^4 = 3/16. Next, we are going to focus on the numerator. Let's multiply out the numerator. When we multiply (x-1)(x-2)(x-3)(3x-4), we are going to get 3x^4, plus lower-degree terms. And, then we have the denominator: (kx-5)^4. We need to expand this, which gives us k4x4 plus lower-degree terms. The key idea here is to only focus on the terms with the highest degree because the other terms will become very small. When x goes to infinity, the other terms are going to be less important than the ones with a higher degree. We can write the expression as follows: lim x→∞: (3x^4 + lower degree terms) / (k4x4 + lower degree terms) = 3/16. In essence, our strategy will be to use the highest power of 'x' in both the numerator and denominator. We will then simplify the fraction to find the limit. This simplified form will enable us to determine the value of 'k'.

Simplifying the Equation: Getting to the Core

Alright, let's roll up our sleeves and get our hands dirty with some algebra. We already know that we only care about the highest-degree terms. The numerator, when fully expanded, will have a leading term of 3x⁴. This is because multiplying the 'x' terms from each factor gives us x * x * x * 3x = 3x⁴. Similarly, in the denominator, expanding (kx - 5)⁴ results in a leading term of k⁴x⁴. This is because (kx)⁴ is k⁴x⁴.

So, we can rewrite the limit as: lim x→∞: (3x⁴) / (k⁴x⁴) = 3/16. You see, guys? The lower-degree terms become insignificant as 'x' heads to infinity. They don't affect the final result. Now, we can cancel out the x⁴ terms, simplifying the equation even further. This leaves us with: 3 / k⁴ = 3/16. This is where the magic happens! We've transformed a complex limit problem into a simple algebraic equation that's much easier to solve. The next step is all about isolating 'k'. To do this, we can cross-multiply: 3 * 16 = 3 * k⁴ which simplifies to 48 = 3k⁴. Dividing both sides by 3 gives us: k⁴ = 16. Finally, we take the fourth root of both sides to find 'k'. Remember, when taking an even root (like the fourth root), we need to consider both positive and negative solutions. So, k = ±2.

Finding the Smallest Value of k: The Final Step

Now we're at the finish line! We've simplified the equation, solved for 'k', and found that k can be either 2 or -2. The question asks us to find the smallest value of 'k'. Between 2 and -2, which is the smallest? Well, obviously, it's -2. Thus, the smallest value of 'k' that satisfies the limit is -2. And just like that, we've solved the problem! See, it wasn't so bad, right?

In Summary: The approach involved simplifying the original expression by focusing on the highest degree terms in the numerator and the denominator. We worked through the limit, expanded and canceled like terms, and then solved for 'k'. In the end, we remembered to find the smallest value from our potential answers, which led us to the correct solution.

Key Takeaways and Tips for Success

Alright, folks, before we sign off, let's recap some critical points to ensure that you master these kinds of problems:

  • Focus on the Highest Degree Terms: When dealing with limits as x approaches infinity, the terms with the highest degree dominate the behavior of the function. Ignore the rest!
  • Simplify, Simplify, Simplify: Always try to simplify the expression before evaluating the limit. This makes it easier to spot patterns and potential cancellations.
  • Remember the Basics: Brush up on your algebra skills, especially factoring and expanding polynomials. These techniques are crucial for simplifying expressions.
  • Consider Both Positive and Negative Solutions: When solving for variables raised to even powers, remember to consider both positive and negative solutions.
  • Practice, Practice, Practice: The more problems you solve, the more comfortable you'll become with the process. Calculus is a skill – and you need to keep practicing to hone it. Try different types of problems and work through them step by step.

I hope you found this breakdown helpful, guys! Keep practicing, and you'll be acing these limit problems in no time. If you have any questions, feel free to ask! Happy calculating!