Limit Of √(3x) - √(x-5) As X→∞: A Step-by-Step Guide

Hey everyone! Today, we're diving headfirst into the fascinating world of limits, specifically tackling the limit of √(3x) - √(x-5) as x zooms off to infinity. This might seem like a daunting mathematical mountain to climb, but fear not! We'll break it down step by step, making sure everyone can follow along. So, grab your thinking caps, and let's get started!

The Challenge: Understanding the Indeterminate Form

Our initial task involves deciphering the intricacies of the limit expression: lim x→∞ (√(3x) - √(x-5)). The first step in evaluating limits often involves direct substitution. However, when we substitute infinity directly into our expression, we encounter a bit of a mathematical puzzle. As x approaches infinity, both √(3x) and √(x-5) also approach infinity. This leaves us with an expression that looks like ∞ - ∞, which, believe it or not, is an indeterminate form. Think of it like a mathematical mystery – we can't immediately tell what the answer is. Indeterminate forms pop up frequently in calculus, and they signal that we need to use a clever trick or technique to unveil the true limit. This ∞ - ∞ situation tells us that we can't just subtract infinities; we need to dig deeper and manipulate the expression to find a determinate form.

To grasp this concept fully, consider that infinity isn't a number but rather a concept representing unbounded growth. Subtracting one infinity from another doesn't necessarily result in zero, as the rates at which these infinities grow might differ. In our case, both square root terms are growing without bound as x increases, but the difference between them isn't immediately clear. This is where the beauty of mathematical techniques comes into play. We'll use algebraic manipulation to rewrite the expression in a way that makes the limit easier to evaluate. The indeterminate form is a clue, a signpost that points us toward the right path. It's a challenge that mathematicians love to tackle, and we're about to join their ranks! So, buckle up as we transform this seemingly complex problem into something much more manageable.

The Solution: Rationalizing the Expression

The key to cracking this limit lies in a technique called rationalizing the expression. Remember those days in algebra class when you had to get rid of square roots in the denominator? We're going to use a similar idea here, but instead of rationalizing a denominator, we'll rationalize the entire expression. This involves multiplying the expression by a clever form of 1 – the conjugate. The conjugate of √(3x) - √(x-5) is √(3x) + √(x-5). By multiplying both the numerator and denominator by this conjugate, we're essentially multiplying by 1, which doesn't change the value of the expression, but it does change its form in a very helpful way.

So, we start by multiplying our original expression by (√(3x) + √(x-5)) / (√(3x) + √(x-5)). When we do this, the numerator transforms from √(3x) - √(x-5) into (√(3x) - √(x-5))(√(3x) + √(x-5)). This is the difference of squares pattern in action! Recall that (a - b)(a + b) = a² - b². Applying this to our numerator, we get (√(3x))² - (√(x-5))², which simplifies to 3x - (x - 5). This is a fantastic step because it eliminates the square roots in the numerator, making it a much simpler expression to deal with. Now, the numerator is just 2x + 5. The denominator, on the other hand, becomes √(3x) + √(x-5). We haven't gotten rid of the square roots entirely, but we've shifted them to the denominator, which turns out to be a strategic move. Rationalizing the expression is like performing a mathematical magic trick – it transforms the problem into a more approachable form. By getting rid of the troublesome square roots in the numerator, we've paved the way for a much cleaner limit evaluation.

Simplifying and Evaluating the Limit

Now that we've rationalized the expression, we have a new form of our limit: lim x→∞ (2x + 5) / (√(3x) + √(x-5)). At first glance, this might still seem a bit intimidating, but we're getting closer! The next step involves simplifying this expression further. A common technique when dealing with limits at infinity involving square roots is to divide both the numerator and the denominator by the highest power of x present in the denominator inside the square root. In this case, that's x. However, because of the square roots, we'll actually divide by √x in the denominator and adjust accordingly in the numerator. To do this correctly, we need to rewrite x in the numerator as √(x²). So, we divide both the numerator and the denominator by √x. This gives us lim x→∞ ( (2x + 5) / √(x²) ) / ( (√(3x) + √(x-5)) / √x ). Breaking this down, we get lim x→∞ ( (2x/√(x²)) + (5/√(x²)) ) / ( (√(3x)/√x) + (√(x-5)/√x) ). Simplifying the square roots, we have lim x→∞ ( 2 + (5/x) ) / ( √3 + √(1 - (5/x)) ).

Now, the magic truly happens! As x approaches infinity, the terms 5/x approach zero. This is a crucial observation. Think about it – as the denominator gets incredibly large, the fraction as a whole shrinks towards zero. This simplifies our limit dramatically. We're left with lim x→∞ (2 + 0) / (√3 + √(1 - 0)), which simplifies to 2 / (√3 + √1) or 2 / (√3 + 1). We've successfully navigated the complexities of infinity and arrived at a concrete expression. To make this answer even cleaner, we can rationalize the denominator again! Multiply both the numerator and denominator by the conjugate of √3 + 1, which is √3 - 1. This gives us (2(√3 - 1)) / ((√3 + 1)(√3 - 1)). Simplifying, we get (2(√3 - 1)) / (3 - 1), which further simplifies to (2(√3 - 1)) / 2. Finally, we arrive at our answer: √3 - 1. So, the limit of √(3x) - √(x-5) as x approaches infinity is √3 - 1. What a journey!

Conclusion: The Power of Mathematical Tools

Wow, we made it! We successfully navigated the world of limits and indeterminate forms, using the power of rationalization and simplification to arrive at our final answer: √3 - 1. This problem showcases the beauty and elegance of mathematical techniques. We started with a seemingly perplexing expression involving infinity, but by applying the right tools, we were able to transform it into something much more manageable and ultimately find a precise value.

The key takeaways from this exploration are the importance of recognizing indeterminate forms, the power of algebraic manipulation (specifically rationalization), and the strategic simplification of expressions when dealing with limits at infinity. Remember, indeterminate forms like ∞ - ∞ aren't roadblocks; they're invitations to apply clever mathematical strategies. Rationalization, as we saw, is a potent weapon in our arsenal, allowing us to eliminate troublesome square roots and reveal the underlying structure of the expression. Dividing by the highest power of x, in this case √x, is another crucial technique for simplifying limits at infinity. It allows us to isolate the dominant terms and see how the expression behaves as x grows without bound.

More broadly, this exercise highlights the problem-solving mindset that's so valuable in mathematics and beyond. When faced with a complex challenge, we don't give up. Instead, we break the problem down into smaller, more manageable steps. We identify the key concepts and techniques that apply. We experiment, we manipulate, and we persevere until we reach a solution. So, the next time you encounter a daunting mathematical problem, remember our journey through this limit. Embrace the challenge, apply the tools you've learned, and unlock the beauty of mathematics! Keep exploring, keep learning, and keep those mathematical muscles flexed!