Mastering Limits At Infinity Solving (3x-1)/√(x^2-6)

Hey guys! Ever found yourself staring at a mathematical expression and wondering what happens as a variable zooms off to infinity? Well, you're not alone! Today, we're going to break down a classic calculus problem that involves finding the limit of a function as x approaches positive infinity. Specifically, we'll tackle this intriguing limit:

$$\lim _{x \rightarrow \infty} \frac{3 x-1}{\sqrt{x^2-6}}$$

Buckle up, because we're about to embark on a mathematical journey that's both enlightening and, dare I say, fun!

Understanding Limits and Infinity

Before we dive into the nitty-gritty of this particular problem, let's take a moment to understand what limits and infinity really mean in the context of calculus. Imagine a function as a machine that takes an input (x) and spits out an output (f(x)). A limit asks: What value does f(x) get closer and closer to as x approaches a certain value (in our case, positive infinity)?

Infinity, denoted by the symbol ∞, isn't a number in the traditional sense. It's a concept that represents something without any bound. When we say x approaches positive infinity, we mean that x is getting larger and larger without any limit. It's like counting upwards forever – you'll never reach the "end," but you'll keep getting closer to the idea of infinity.

Now, let's talk about why we care about limits as x approaches infinity. In many real-world situations, we want to know the long-term behavior of a system. For example, if we're modeling the population growth of a species, we might be interested in what happens to the population as time goes to infinity. Or, in physics, we might want to know the terminal velocity of an object falling through the air – that is, the velocity it approaches as time goes on forever. These are just a couple of the many reasons why understanding limits at infinity is crucial.

Key Concepts to Remember

To successfully navigate limits at infinity, there are a few key concepts that are important to keep in mind. These concepts form the foundation for our problem-solving strategy, so make sure you've got a good grasp of them before we move on.

1. Dominant Terms: When dealing with rational functions (polynomials divided by polynomials) as x approaches infinity, the dominant terms are the terms with the highest powers of x. These terms dictate the overall behavior of the function as x gets extremely large. Lower-power terms become insignificant in comparison.
2. Dividing by the Highest Power: A common technique for evaluating limits at infinity is to divide both the numerator and the denominator of a rational function by the highest power of x that appears in the denominator. This simplifies the expression and often allows us to directly evaluate the limit.
3. Limits of 1/x^n: A crucial fact to remember is that for any positive integer n, the limit of 1/x^n as x approaches infinity is always zero. This is because as x gets larger and larger, 1/x^n gets smaller and smaller, approaching zero.
4. The Squeeze Theorem (Optional): While not directly used in this problem, the Squeeze Theorem is a powerful tool for evaluating limits. It states that if a function is "squeezed" between two other functions that have the same limit, then the function in the middle must also have that limit.

With these concepts in mind, we're well-equipped to tackle our limit problem. Let's move on to the exciting part – solving it!

Solving the Limit Problem Step-by-Step

Okay, let's get down to business and solve the limit:

$$\lim _{x \rightarrow \infty} \frac{3 x-1}{\sqrt{x^2-6}}$$

Step 1: Identify the Highest Power of x in the Denominator

First things first, we need to figure out the highest power of x lurking in the denominator. Notice that we have a square root. Inside the square root, we have x^2. When we take the square root of x^2, we effectively get |x|. However, since x is approaching positive infinity, we can safely assume x is positive, so √(x^2) = x. Therefore, the highest power of x in the denominator is simply x.

Step 2: Divide Numerator and Denominator by the Highest Power

This is the key step! We're going to divide both the numerator and the denominator by x. This might seem a bit tricky at first, especially with the square root in the denominator, but stick with me. Remember, to bring x inside the square root, we need to square it.

So, let's rewrite the expression:

$$\lim _{x \rightarrow \infty} \frac{\frac{3 x-1}{x}}{\frac{\sqrt{x^2-6}}{x}}$$

Now, let's simplify the numerator and denominator separately.

• Numerator:

  $$\frac{3 x-1}{x} = \frac{3x}{x} - \frac{1}{x} = 3 - \frac{1}{x}$$

• Denominator:

  Here's where the square root magic happens. We bring the x inside the square root by squaring it:

  $$\frac{\sqrt{x^2-6}}{x} = \frac{\sqrt{x^2-6}}{\sqrt{x^2}} = \sqrt{\frac{x^2-6}{x^2}}$$

  Now, we simplify the expression inside the square root:

  $$\sqrt{\frac{x^2-6}{x^2}} = \sqrt{\frac{x^2}{x^2} - \frac{6}{x^2}} = \sqrt{1 - \frac{6}{x^2}}$$

Step 3: Rewrite the Limit

Now that we've simplified the numerator and denominator, let's rewrite the limit expression:

$$\lim _{x \rightarrow \infty} \frac{3 - \frac{1}{x}}{\sqrt{1 - \frac{6}{x^2}}}$$

Step 4: Evaluate the Limit

This is the moment of truth! We're going to use our knowledge of limits to evaluate the expression as x approaches infinity.

Remember that the limit of 1/x^n as x approaches infinity is 0. So, both 1/x and 6/x^2 will approach 0 as x goes to infinity.

Therefore, we have:

$$\lim _{x \rightarrow \infty} \frac{3 - \frac{1}{x}}{\sqrt{1 - \frac{6}{x^2}}} = \frac{3 - 0}{\sqrt{1 - 0}} = \frac{3}{\sqrt{1}} = \frac{3}{1} = 3$$

Step 5: State the Final Answer

Tada! We've found the limit! The limit of the function as x approaches positive infinity is 3.

$$\lim _{x \rightarrow \infty} \frac{3 x-1}{\sqrt{x^2-6}} = 3$$

Alternative Approaches and Insights

While we've successfully solved the limit using the divide-by-the-highest-power method, it's always good to explore alternative approaches and gain deeper insights into the problem.

Thinking About Dominant Terms

Let's revisit the concept of dominant terms. As x becomes extremely large, the -1 in the numerator becomes insignificant compared to 3x. Similarly, the -6 inside the square root in the denominator becomes insignificant compared to x^2. So, for very large values of x, the function behaves approximately like:

$$\frac{3x}{\sqrt{x^2}} = \frac{3x}{|x|}$$

Since we're considering x approaching positive infinity, |x| is simply x. So, the function simplifies to:

$$\frac{3x}{x} = 3$$

This gives us a quick intuitive understanding of why the limit is 3. The dominant terms essentially "cancel out," leaving us with a constant.

L'Hôpital's Rule (Advanced)

For those familiar with L'Hôpital's Rule, this is another powerful tool for evaluating limits of indeterminate forms (like ∞/∞). L'Hôpital's Rule states that if the limit of f(x)/g(x) as x approaches a value is of the form 0/0 or ∞/∞, then:

$$\lim _{x \rightarrow c} \frac{f(x)}{g(x)} = \lim _{x \rightarrow c} \frac{f'(x)}{g'(x)}$$

where f'(x) and g'(x) are the derivatives of f(x) and g(x), respectively.

To apply L'Hôpital's Rule to our problem, we first need to verify that the limit is of the form ∞/∞. As x approaches infinity, both the numerator (3x - 1) and the denominator (√(x^2 - 6)) approach infinity. So, we can apply L'Hôpital's Rule.

Let's find the derivatives:

• Numerator: f(x) = 3x - 1, so f'(x) = 3
• Denominator: g(x) = √(x^2 - 6), so g'(x) = (1/2)(x^2 - 6)^(-1/2) * (2x) = x/√(x^2 - 6)

Now, let's rewrite the limit using L'Hôpital's Rule:

$$\lim _{x \rightarrow \infty} \frac{3}{\frac{x}{\sqrt{x^2-6}}} = \lim _{x \rightarrow \infty} \frac{3\sqrt{x^2-6}}{x}$$

This looks familiar! It's very similar to our original problem. We can use the same divide-by-the-highest-power technique we used before to solve this limit, and we'll arrive at the same answer: 3.

While L'Hôpital's Rule can be a powerful tool, it's important to note that it's not always the most efficient method. In this case, the divide-by-the-highest-power method is arguably simpler and more direct.

Real-World Applications and Significance

Okay, so we've conquered this limit problem. But you might be thinking, "Why should I care about this in the real world?" Well, the concept of limits at infinity has numerous applications in various fields.

• Physics: As mentioned earlier, limits at infinity are used to determine terminal velocities, analyze the behavior of systems over long periods, and model the decay of radioactive materials.
• Engineering: Engineers use limits to analyze the stability of systems, design control systems, and optimize processes.
• Economics: Economists use limits to model long-term economic trends, analyze market behavior, and predict financial outcomes.
• Computer Science: Limits are used in algorithm analysis to determine the efficiency of algorithms as the input size grows. They're also used in numerical analysis to approximate solutions to equations.

In essence, understanding limits at infinity allows us to make predictions about the long-term behavior of systems and processes. It's a fundamental concept in calculus and a powerful tool for solving real-world problems.

Conclusion: Mastering Limits at Infinity

So, there you have it, guys! We've successfully navigated the world of limits at infinity, tackled a challenging problem step-by-step, and explored alternative approaches and real-world applications. Hopefully, you now have a solid understanding of how to find the limit of a function as x approaches positive infinity.

Remember the key concepts: dominant terms, dividing by the highest power, and the behavior of 1/x^n as x goes to infinity. Practice these techniques, and you'll be well-equipped to conquer any limit problem that comes your way!

Calculus can seem daunting at first, but with a little practice and a lot of curiosity, you can unlock its power and beauty. Keep exploring, keep questioning, and keep learning! And who knows, maybe you'll be the one to discover the next groundbreaking application of limits at infinity. Cheers to your mathematical journey!