Solving Limits At Infinity A Step-by-Step Guide For 3x - √(9x² + 2x - 7)

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Hey guys! Ever stumbled upon a limit problem that looks like it's from another dimension? You know, the kind where 'x' is trying to become infinity? Well, today we're diving deep into one of those – specifically, how to solve the limit of 3x - √(9x² + 2x - 7) as x approaches infinity. This type of problem is super common in calculus, especially when you're prepping for exams like the SBMPTN. So, let's break it down step-by-step, make it less scary, and more 'I got this!'

Understanding Limits Approaching Infinity

Before we jump into the nitty-gritty of this particular problem, let's get our heads around what limits approaching infinity actually mean. When we say "limit of a function as x approaches infinity," we're asking: What value does the function get closer and closer to as x gets super, super big? Think of it like a race where x is running towards infinity – we want to know where the function is headed as x sprints towards the finish line. The crucial thing here is that infinity isn't a number; it's a concept. It represents something without any bound. So, we're not plugging infinity into the function; we're observing the function's behavior as x grows without limit. When dealing with limits at infinity, identifying the highest power of x is crucial because it dictates the function's end behavior. In our problem, the highest power of x inside the square root is x², but since it's under a square root, it effectively behaves like x. This means we're comparing terms of similar magnitude, which is why we can't simply ignore terms as x approaches infinity. We need to manipulate the expression algebraically to reveal the true limit. Techniques such as multiplying by the conjugate or dividing by the highest power of x are commonly used to simplify these expressions and make the limit calculation straightforward.

Now, when we're faced with expressions involving square roots, things can get a little trickier. We can't just plug in infinity and call it a day because we might end up with indeterminate forms like infinity minus infinity. That's where the fun begins! To tackle these problems, we often need to use some algebraic tricks to massage the expression into a form that's easier to handle. One of the most common techniques is to multiply by the conjugate. This might sound like some fancy math jargon, but it's actually quite straightforward. The conjugate is simply the same expression but with the sign in the middle flipped. For example, the conjugate of a - b is a + b. Multiplying by the conjugate helps us get rid of the square root and simplify the expression. Another important concept to keep in mind is the dominant term. As x approaches infinity, some terms in the expression become much larger than others. The dominant term is the one that grows the fastest. Identifying the dominant term can help us simplify the problem and focus on what really matters. In our case, both 3x and √(9x²) are dominant terms, and they both grow linearly with x. This is why we need to be careful when evaluating the limit and can't just ignore the other terms. We need to manipulate the expression to see how these dominant terms interact with each other and determine the overall behavior of the function as x approaches infinity.

The Problem: limit x approaches infinity of 3x - √(9x² + 2x - 7)

Let's restate the problem we're tackling today: We need to find the limit of 3x - √(9x² + 2x - 7) as x approaches infinity. At first glance, this might look intimidating. We've got a square root, a subtraction, and infinity all thrown into the mix. But don't worry, we're going to break it down step by step. The core issue here is the subtraction of two terms that both grow infinitely large as x approaches infinity. This creates an indeterminate form of the type infinity minus infinity, which doesn't immediately tell us the limit's value. We can't simply subtract infinity from infinity; we need a more rigorous approach to determine the expression's behavior. The presence of the square root further complicates matters, as it prevents us from directly combining terms. To overcome this, we'll employ a common technique for dealing with such limits: multiplying by the conjugate. This involves multiplying both the numerator and denominator of the expression by the conjugate of the term involving the square root. The conjugate will help us eliminate the square root and simplify the expression into a form where the limit can be more easily evaluated. This technique is particularly useful when dealing with expressions involving square roots and indeterminate forms, as it allows us to manipulate the expression algebraically and reveal the underlying behavior as x approaches infinity.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this limit problem! Here’s how we can tackle it:

Step 1: Multiply by the Conjugate

As we discussed, the first step is to multiply our expression by its conjugate. The conjugate of 3x - √(9x² + 2x - 7) is 3x + √(9x² + 2x - 7). We multiply both the numerator and denominator by this conjugate. Remember, multiplying by the conjugate is like multiplying by 1, so we're not changing the value of the expression, just its form. By multiplying by the conjugate, we aim to eliminate the square root and simplify the expression. This technique is a common trick in calculus for dealing with limits involving square roots and indeterminate forms. The conjugate has the same terms but with the opposite sign between them, which allows us to use the difference of squares identity to get rid of the square root. When we multiply the original expression by its conjugate, the square root term gets squared, effectively removing the square root. This simplifies the expression and makes it easier to evaluate the limit. So, multiplying by the conjugate is a crucial first step in solving this problem, as it sets the stage for further simplification and ultimately leads us to the solution. Let's take a closer look at how this multiplication works and what the resulting expression looks like.

So, we have:

lim (x→∞) [3x - √(9x² + 2x - 7)] * [3x + √(9x² + 2x - 7)] / [3x + √(9x² + 2x - 7)]

Step 2: Simplify the Expression

Now, let's simplify the numerator. We'll use the difference of squares formula: (a - b)(a + b) = a² - b². This formula is our best friend when dealing with conjugates. It allows us to eliminate the square root and simplify the expression into a form that's easier to work with. The difference of squares formula is a fundamental algebraic identity that pops up in various mathematical contexts, including calculus. It's particularly useful when we want to get rid of square roots, as it transforms a product of two binomials into a difference of two squares. In our case, applying this formula will eliminate the square root in the numerator and leave us with a simpler expression. This simplification is crucial because it allows us to combine like terms and rewrite the expression in a form where we can evaluate the limit as x approaches infinity. So, mastering the difference of squares formula is a key skill for solving limit problems involving conjugates and square roots. Let's apply this formula to our expression and see how it simplifies.

Applying this, we get:

lim (x→∞) [(3x)² - (√(9x² + 2x - 7))²] / [3x + √(9x² + 2x - 7)]

Which simplifies to:

lim (x→∞) [9x² - (9x² + 2x - 7)] / [3x + √(9x² + 2x - 7)]

Further simplifying the numerator, we have:

lim (x→∞) [-2x + 7] / [3x + √(9x² + 2x - 7)]

Step 3: Divide by the Highest Power of x

The next step is to divide both the numerator and the denominator by the highest power of x present in the expression. In this case, the highest power of x is x (or √x² under the square root). Dividing by the highest power of x is a common technique for evaluating limits at infinity. It helps us identify the dominant terms in the expression and determine the limit's value as x approaches infinity. When we divide both the numerator and the denominator by the highest power of x, we're essentially normalizing the expression. This means we're rewriting the expression in terms of ratios of lower powers of x, which makes it easier to see how the expression behaves as x becomes very large. For example, terms with lower powers of x will approach zero, while terms with the same power of x will approach a constant. This technique allows us to focus on the terms that matter most as x goes to infinity and ignore the ones that become negligible. So, dividing by the highest power of x is a crucial step in evaluating limits at infinity, and it's a technique that you'll use frequently in calculus. Let's apply this to our expression and see how it simplifies further.

So, we divide both the numerator and the denominator by x:

lim (x→∞) [-2 + 7/x] / [3 + √(9x² + 2x - 7)/x]

To bring x inside the square root, we need to square it:

lim (x→∞) [-2 + 7/x] / [3 + √(9 + 2/x - 7/x²)]

Step 4: Evaluate the Limit

Now, we can evaluate the limit as x approaches infinity. Remember that as x gets super big, terms like 7/x, 2/x, and 7/x² will approach zero. This is a key concept in evaluating limits at infinity. As x grows without bound, any constant divided by x will approach zero. This is because the denominator becomes infinitely large, while the numerator remains constant. This means that the overall fraction becomes infinitesimally small, effectively approaching zero. This principle allows us to simplify expressions and eliminate terms that become negligible as x approaches infinity. So, when we see terms like 7/x, 2/x, or 7/x² in a limit problem, we know that these terms will vanish as x goes to infinity. This makes it much easier to evaluate the limit and determine the function's behavior as x becomes very large. Let's apply this concept to our expression and see what the limit turns out to be.

So, we have:

lim (x→∞) [-2 + 0] / [3 + √(9 + 0 - 0)]

Which simplifies to:

-2 / [3 + √9]

-2 / [3 + 3]

-2 / 6

-1/3

Final Answer

So, the limit of 3x - √(9x² + 2x - 7) as x approaches infinity is -1/3. We did it! Wasn't that fun? (Okay, maybe 'fun' is a strong word, but you get the idea!). To summarize, we tackled this problem by first recognizing the indeterminate form and then employing the strategy of multiplying by the conjugate. This allowed us to eliminate the square root and simplify the expression. We then divided by the highest power of x to further simplify and identify the dominant terms. Finally, we evaluated the limit by letting x approach infinity and recognizing that terms with x in the denominator approach zero. This step-by-step approach is a powerful tool for solving a wide range of limit problems, and mastering it will significantly boost your calculus skills. Remember, practice makes perfect, so try applying this technique to similar problems to solidify your understanding. And don't be afraid to break down complex problems into smaller, more manageable steps. With a little patience and persistence, you can conquer any limit problem that comes your way.

Key Takeaways for Limit Problems

Before we wrap up, let's quickly recap the key takeaways from this problem. These tips will come in handy when you're tackling similar limit problems in the future:

  1. Identify the Indeterminate Form: The very first step in solving any limit problem, especially those approaching infinity, is to recognize if you have an indeterminate form. Common indeterminate forms include ∞ - ∞, ∞/∞, 0/0, etc. Spotting these forms tells you that you can't just plug in infinity and call it a day. You need to do some algebraic manipulation first. Recognizing the indeterminate form is like diagnosing the problem before prescribing a solution. It helps you understand what kind of techniques you'll need to use to simplify the expression and evaluate the limit. For example, if you see ∞ - ∞, you might think about multiplying by the conjugate. If you see ∞/∞, you might consider dividing by the highest power of x. So, always start by identifying the indeterminate form to guide your approach.

  2. Multiply by the Conjugate: When dealing with square roots and subtraction (or addition), multiplying by the conjugate is often your best friend. This technique helps you eliminate the square root and simplify the expression. Remember, the conjugate is the same expression with the sign flipped. Multiplying by the conjugate is like using a special key to unlock a hidden structure in the expression. It allows you to rewrite the expression in a more manageable form where you can combine like terms and identify the dominant behavior as x approaches infinity. This technique is particularly useful when dealing with indeterminate forms like ∞ - ∞, as it helps you transform the expression into a form where you can apply other simplification techniques.

  3. Divide by the Highest Power of x: For limits approaching infinity, dividing both the numerator and denominator by the highest power of x is a powerful technique. This helps you identify the dominant terms and simplify the expression as x becomes very large. Dividing by the highest power of x is like zooming out to see the bigger picture. It allows you to ignore the terms that become negligible as x approaches infinity and focus on the ones that truly determine the limit's value. This technique is particularly useful when dealing with rational functions, where you have polynomials in both the numerator and the denominator. By dividing by the highest power of x, you can easily see which terms dominate and how the expression behaves as x becomes very large.

  4. Evaluate Carefully: After simplifying, take a deep breath and carefully evaluate the limit. Remember that terms like 1/x, 1/x², etc., will approach zero as x approaches infinity. Evaluating carefully means paying attention to the details and making sure you're applying the limit rules correctly. It's easy to make a small mistake, especially when dealing with complex expressions, so take your time and double-check your work. Remember that the goal is to determine the function's behavior as x becomes very large, so focus on the terms that will dominate and how they interact with each other. And don't be afraid to break down the evaluation into smaller steps to make sure you're not missing anything.

By keeping these takeaways in mind and practicing regularly, you'll become a limit-solving pro in no time. Good luck with your studies, and remember, math is challenging, but it's also super rewarding when you finally crack a tough problem!

Practice Problems

Want to put your newfound skills to the test? Here are a couple of practice problems you can try:

  1. lim (x→∞) [√(x² + 4x) - x]
  2. lim (x→∞) [2x - √(4x² - 5x + 1)]

Try applying the steps we discussed in this article, and you'll be well on your way to solving these problems. Remember, the key is to break down the problem into smaller steps, identify the indeterminate form, and use the appropriate algebraic techniques to simplify the expression. And don't be afraid to ask for help if you get stuck! There are plenty of resources available online and in textbooks to help you master calculus. Good luck, and happy problem-solving!

That's all for today, guys! Keep practicing, and you'll ace those limit problems in no time. Happy studying!