Calculating Lim X→0 Of 2x Ln(x) Using L'Hôpital's Rule
Hey guys! Let's dive into a classic calculus problem: finding the limit of 2x * ln(x) as x approaches 0. This might seem tricky at first, but we've got a powerful tool in our arsenal: L'Hôpital's Rule. We'll break down the steps, understand how to apply the rule correctly, and nail down the final answer. So, grab your thinking caps, and let's get started!
Understanding the Indeterminate Form
Before we even think about L'Hôpital's Rule, it's crucial to understand why we need it in the first place. If we directly substitute x = 0 into the expression 2x * ln(x), we run into a bit of a problem. We get 2 * 0 * ln(0), which translates to 0 * (-∞). This is what we call an indeterminate form. It doesn't immediately tell us what the limit is. Think of it like a mathematical mystery that we need to solve.
Indeterminate forms like 0 * (±∞), 0/0, ∞/∞, ∞ - ∞, 1^∞, 0^0, and ∞^0 signal that we can't just plug in the value and call it a day. We need a clever technique to unravel the limit's true value. That's where L'Hôpital's Rule comes into play. L'Hôpital's Rule is a mathematical superhero that swoops in to save the day when we encounter these indeterminate forms. It provides a method for evaluating limits of fractions where both the numerator and denominator approach either 0 or infinity. However, it's crucial to remember that L'Hôpital's Rule isn't a magic bullet for all limits. It's specifically designed for indeterminate forms like 0/0 and ∞/∞. Applying it to other situations can lead to incorrect results. The beauty of L'Hôpital's Rule lies in its ability to transform a seemingly unsolvable limit into a more manageable one. By taking the derivatives of the numerator and the denominator, we often simplify the expression and reveal the limit's true value. But, as with any powerful tool, it's essential to wield it correctly. Understanding the conditions under which L'Hôpital's Rule applies is paramount to avoid misapplication and ensure accurate results. So, before we jump into applying the rule, let's make sure we've correctly identified the indeterminate form and that L'Hôpital's Rule is indeed the appropriate technique for the task at hand. This careful preparation will pave the way for a successful and accurate limit evaluation.
Preparing for L'Hôpital's Rule: Transforming the Expression
The key to using L'Hôpital's Rule is to have a fraction where both the numerator and denominator approach either 0 or infinity. Our current expression, 2x * ln(x), isn't in that form. So, we need to do a little mathematical maneuvering to get it ready for the rule. The trick here is to rewrite the expression as a fraction. We can do this by moving the 2x term to the denominator as its reciprocal. Think of it like this: multiplying by 2x is the same as dividing by 1/(2x). So, we can rewrite 2x * ln(x) as ln(x) / (1/(2x)).
Now, let's see what happens as x approaches 0. The numerator, ln(x), approaches negative infinity (-∞). The denominator, 1/(2x), approaches infinity (∞). Aha! We've successfully transformed our expression into the indeterminate form (-∞)/∞. This is one of the forms where L'Hôpital's Rule can be applied. This transformation is a crucial step in the process. It's like preparing the ingredients before you start cooking – you need to have everything in the right form before you can create the final dish. Without this step, we wouldn't be able to use L'Hôpital's Rule, and we'd be stuck with our indeterminate form. The clever manipulation of rewriting the expression as a fraction allows us to bring L'Hôpital's Rule into the picture. It's a testament to the power of algebraic manipulation in solving calculus problems. By recognizing the need for this transformation, we've opened the door to a solution that was previously hidden. So, with our expression now in the appropriate form, we're ready to move on to the next step: applying L'Hôpital's Rule itself. But before we do that, let's take a moment to appreciate the elegance of this transformation. It's a prime example of how a little mathematical creativity can unlock the solution to a seemingly difficult problem. And that, my friends, is one of the many reasons why math can be so darn fascinating!
Applying L'Hôpital's Rule: Taking Derivatives
Alright, we've got our expression in the right form, so it's time to unleash L'Hôpital's Rule! Remember, this rule tells us that if the limit of f(x)/g(x) as x approaches a is in an indeterminate form (0/0 or ∞/∞), then the limit is the same as the limit of f'(x)/g'(x), where f'(x) and g'(x) are the derivatives of f(x) and g(x), respectively. In our case, f(x) = ln(x) and g(x) = 1/(2x).
So, let's find those derivatives. The derivative of ln(x) is simply 1/x. Now, for the derivative of 1/(2x), we can rewrite it as (1/2)x^(-1). Using the power rule, the derivative becomes (1/2)*(-1)x^(-2), which simplifies to -1/(2x^2). Now we have all the pieces we need. We can apply L'Hôpital's Rule and say that the limit of ln(x) / (1/(2x)) as x approaches 0 is the same as the limit of (1/x) / (-1/(2x^2)) as x approaches 0. But we're not done yet! We've taken the derivatives, but we still need to simplify the expression before we can evaluate the limit. The process of taking derivatives is at the heart of L'Hôpital's Rule. It's the mechanism by which we transform the original indeterminate form into a potentially simpler expression. However, the derivatives themselves are just one piece of the puzzle. The real magic happens when we simplify and evaluate the resulting limit. Taking derivatives can sometimes feel like a mechanical process, but it's crucial to remember the underlying concept. The derivative represents the instantaneous rate of change of a function. By taking the derivatives of the numerator and denominator, we're essentially comparing their rates of change as x approaches a certain value. This comparison is what allows us to unravel the indeterminate form and find the true limit. So, as we move forward with the problem, let's keep in mind the significance of these derivatives. They're not just mathematical symbols; they're the key to unlocking the limit's hidden value. And with our derivatives in hand, we're one step closer to cracking the code and finding the solution.
Simplifying the Expression: A Little Algebra Magic
We've applied L'Hôpital's Rule and found the derivatives, but we're not quite at the finish line yet. Our expression now looks like (1/x) / (-1/(2x^2)). This looks a bit messy, so let's simplify it using some good old-fashioned algebra. Remember, dividing by a fraction is the same as multiplying by its reciprocal. So, we can rewrite (1/x) / (-1/(2x^2)) as (1/x) * (-2x^2/1). Now, we can simplify by canceling out an x term. This gives us -2x. Ah, much better! Now we have a much simpler expression to work with. This simplification step is often overlooked, but it's incredibly important. It's like cleaning up your workspace before you start a new task – it makes everything easier and more efficient. Without simplification, we might end up with a complex expression that's difficult to evaluate. But with a little algebraic magic, we've transformed our messy fraction into a clean and simple term. The beauty of simplification lies in its ability to reveal the underlying structure of an expression. By canceling out common factors and combining like terms, we strip away the unnecessary complexity and expose the core relationship. In this case, simplification has transformed a complicated fraction into a simple linear term. This makes the final evaluation of the limit much easier. So, let's not underestimate the power of simplification. It's a valuable tool in our mathematical arsenal, and it can often make the difference between a difficult problem and an easy one. And with our expression now simplified, we're ready to take the final step: evaluating the limit.
Evaluating the Limit: The Grand Finale
Okay, we've done the hard work. We transformed the expression, applied L'Hôpital's Rule, and simplified the result. Now comes the moment of truth: evaluating the limit. Our simplified expression is -2x. So, we need to find the limit of -2x as x approaches 0. This is actually quite straightforward. As x gets closer and closer to 0, -2x also gets closer and closer to 0. Therefore, the limit of -2x as x approaches 0 is 0. And there you have it! We've successfully navigated the indeterminate form and found the limit. The limit of 2x * ln(x) as x approaches 0 is 0. This final evaluation is the culmination of all our efforts. It's the moment where we reap the rewards of our hard work and see the answer revealed. After all the transformations, derivatives, and simplifications, we've arrived at a clear and concise result. The satisfaction of solving a limit problem like this is immense. It's like completing a puzzle, where each step fits perfectly into place to create the final picture. The evaluation of the limit is not just a mechanical step; it's the realization of the underlying mathematical concept. It's the moment where we see how the function behaves as x approaches a certain value. In this case, we've discovered that even though the expression initially presented an indeterminate form, the function actually converges to a finite value as x approaches 0. And that, my friends, is the power of calculus. It allows us to explore the behavior of functions in situations where ordinary algebra falls short. So, let's take a moment to celebrate our success. We've conquered a challenging limit problem, and we've learned valuable techniques along the way. And with this newfound knowledge, we're ready to tackle even more complex mathematical mysteries.
Choosing the Correct Answer
We've meticulously calculated the limit of 2x * ln(x) as x approaches 0, and we've arrived at the answer: 0. Now, let's take a look at the multiple-choice options provided: A) O limite é 0 B) O limite é [Other options]. Based on our calculations, the correct answer is clearly A) O limite é 0. We've successfully navigated the indeterminate form, applied L'Hôpital's Rule correctly, and simplified the expression to arrive at the accurate result. This final step of choosing the correct answer is a crucial part of the problem-solving process. It's not enough to simply perform the calculations; we must also be able to interpret the results and select the appropriate answer from the given options. In this case, the answer was straightforward and aligned perfectly with our calculations. But in other situations, the options might be more nuanced, requiring a careful consideration of the context and the implications of our results. So, let's always remember to double-check our work and ensure that we're selecting the answer that best reflects our understanding of the problem. And with the correct answer chosen, we can confidently say that we've mastered this limit problem and are ready to move on to new mathematical challenges.
In conclusion, we've successfully calculated the limit of 2x * ln(x) as x approaches 0 using L'Hôpital's Rule. We transformed the expression, took derivatives, simplified, and evaluated the limit, arriving at the answer 0. The correct option is A) O limite é 0. Great job, guys! You've tackled a classic calculus problem and come out victorious. Keep up the awesome work, and remember, math can be fun!
