Limit Of (√(1 + X) - √(1 - X)) / X As X Approaches 1 A Calculus Problem
Hey guys! Today, we're diving deep into the fascinating world of calculus to tackle a tricky limit problem. We're going to explore the limit of the expression (√(1 + x) - √(1 - x)) / x as x approaches 1. This problem might seem daunting at first, but with a few clever techniques and a dash of algebraic manipulation, we'll crack it open and reveal the answer. So, buckle up and let's embark on this mathematical journey together!
The Problem at Hand
Before we jump into the solution, let's clearly state the problem we're trying to solve. We want to find the limit of the following expression:
lim (x→1) [ (√(1 + x) - √(1 - x)) / x ]
where:
lim (x→1)means we're looking at what happens to the expression as x gets closer and closer to 1.√denotes the square root.
The Challenge: Our main challenge here is that if we directly substitute x = 1 into the expression, we encounter a pesky 0 in the denominator, which is a big no-no in the math world. This is what we call an indeterminate form, and it means we need to employ some clever techniques to evaluate the limit.
The Answer Options: We've been given a set of possible answers to choose from:
A) 0
B) 1
C) 2
D) Infinito (Infinity)
E) Não existe limite (Limit does not exist)
Our goal is to figure out which of these options is the correct limit, and more importantly, to understand why it's the correct answer. So, let's roll up our sleeves and get to work!
Initial Assessment: Why Direct Substitution Fails
The first thing we always try when evaluating a limit is direct substitution – simply plugging in the value that x is approaching. In this case, that means substituting x = 1 into our expression:
(√(1 + 1) - √(1 - 1)) / 1 = (√2 - √0) / 1 = √2 / 1 = √2
Wait a minute! This seems straightforward, but there's a subtle issue. When x approaches 1, the denominator approaches 0. So, let's re-evaluate by directly substituting x = 1, we get:
(√(1 + 1) - √(1 - 1)) / 1 = (√2 - 0) / 1 = √2
However, this initial attempt glosses over a crucial detail. As x gets closer and closer to 1, both the numerator and the denominator are changing. We can't simply treat them as static values. The expression takes the form of 0/0, which is an indeterminate form. This means that the limit could be anything – it could be 0, it could be infinity, or it could be some finite number. We need a more sophisticated approach to figure it out. The indeterminate form signals that we need to manipulate the expression algebraically to reveal the true behavior of the limit.
The Conjugate Multiplication Technique
Alright, guys, now we're getting to the good stuff! Since direct substitution failed us, we need to pull out a more powerful tool from our calculus toolbox. In this case, the perfect tool for the job is the conjugate multiplication technique. This technique is particularly useful when we're dealing with expressions involving square roots.
What is a Conjugate? The conjugate of an expression like (√(1 + x) - √(1 - x)) is simply (√(1 + x) + √(1 - x)). We change the minus sign in the middle to a plus sign. It might seem like a simple trick, but it's incredibly effective.
Why Use the Conjugate? The magic of the conjugate lies in what happens when we multiply an expression by its conjugate. Remember the difference of squares formula: (a - b)(a + b) = a² - b²? This is exactly what we're going to exploit. Multiplying by the conjugate will help us get rid of the square roots in the numerator, making the expression much easier to work with.
Applying the Technique: Let's multiply both the numerator and the denominator of our expression by the conjugate of the numerator:
lim (x→1) [ (√(1 + x) - √(1 - x)) / x ] * [ (√(1 + x) + √(1 - x)) / (√(1 + x) + √(1 - x)) ]
Notice that we're multiplying by a fraction that's equal to 1, so we're not changing the value of the expression. We're just changing its form.
Simplifying the Expression
Okay, guys, now comes the fun part – simplifying the expression we obtained after multiplying by the conjugate. Let's focus on the numerator first. Using the difference of squares formula, we have:
(√(1 + x) - √(1 - x)) (√(1 + x) + √(1 - x)) = (√(1 + x))² - (√(1 - x))² = (1 + x) - (1 - x) = 1 + x - 1 + x = 2x
See how the square roots magically disappeared? That's the power of the conjugate at work!
Now, let's rewrite our limit expression with the simplified numerator:
lim (x→1) [ 2x / (x * (√(1 + x) + √(1 - x))) ]
Notice that we have an x in both the numerator and the denominator. We can cancel them out, which further simplifies our expression:
lim (x→1) [ 2 / (√(1 + x) + √(1 - x)) ]
We've successfully transformed our original expression into a much simpler form. The indeterminate form is gone, and we're now in a position to directly evaluate the limit.
Evaluating the Limit After Simplification
Alright, folks, we've done the hard work of simplifying the expression. Now, let's reap the rewards and finally evaluate the limit. We're now looking at:
lim (x→1) [ 2 / (√(1 + x) + √(1 - x)) ]
We can now safely use direct substitution. Let's plug in x = 1:
2 / (√(1 + 1) + √(1 - 1)) = 2 / (√2 + √0) = 2 / √2
To simplify this further, we can rationalize the denominator by multiplying both the numerator and the denominator by √2:
(2 / √2) * (√2 / √2) = 2√2 / 2 = √2
So, we've arrived at our limit: √2. But wait! Looking back at our answer options, we don't see √2 listed. This means we need to re-examine our steps. Did we make a mistake somewhere?
Let's carefully go through the conjugate multiplication and simplification again.
lim (x→1) [ (√(1 + x) - √(1 - x)) / x ] * [ (√(1 + x) + √(1 - x)) / (√(1 + x) + √(1 - x)) ]
lim (x→1) [ ((1 + x) - (1 - x)) / (x * (√(1 + x) + √(1 - x))) ]
lim (x→1) [ (2x) / (x * (√(1 + x) + √(1 - x))) ]
lim (x→1) [ 2 / (√(1 + x) + √(1 - x)) ]
Substituting x = 1:
2 / (√(1 + 1) + √(1 - 1)) = 2 / (√2 + 0) = 2 / √2 = √2
It seems we haven't made any errors in our calculations. However, we still don't see √2 among the answer options. This suggests there might be a mistake in the provided answer choices or the problem statement itself. But let's not jump to conclusions just yet.
We need to double-check if the question was copied correctly or if we misunderstood something. Let's rewind and make sure we're solving the right problem and interpreting it accurately. Maybe we missed a crucial detail or made an assumption that wasn't valid. It's essential to have a critical eye and question everything, even our own work! If, after careful review, we still arrive at √2 and the options remain as they are, we can confidently state that there's likely an issue with the problem itself.
Final Thoughts and Key Takeaways
Okay, guys, we've taken a deep dive into this limit problem, and it's been quite the adventure! We started with a seemingly simple expression, but we quickly encountered the dreaded indeterminate form. This led us to employ a powerful technique – conjugate multiplication – to simplify the expression and reveal its true behavior as x approaches 1.
Key Takeaways:
- Direct Substitution: Always try direct substitution first. It's the simplest approach, and sometimes it works! But be wary of indeterminate forms.
- Indeterminate Forms: When you encounter 0/0, ∞/∞, or other indeterminate forms, it means you need to do more work. Algebraic manipulation is often the key.
- Conjugate Multiplication: This is a powerful technique for dealing with expressions involving square roots. It helps eliminate the square roots and simplify the expression.
- Simplify, Simplify, Simplify: Always try to simplify your expression as much as possible before evaluating the limit. This makes the problem much easier to handle.
- Double-Check Your Work: It's crucial to carefully review your steps and make sure you haven't made any mistakes, especially when dealing with complex problems.
- Critical Thinking: Don't be afraid to question the problem itself. If your answer doesn't match the options, it could be a sign of an error in the problem statement or the answer choices.
In conclusion, while we arrived at a solution of √2, which doesn't match the provided options, the process we followed highlights the core principles of limit evaluation. Keep these techniques in your mathematical toolkit, and you'll be well-equipped to tackle a wide range of limit problems!
If the answer options were different, and √2 was present, we could confidently select it as the correct limit. However, in this specific scenario, we need to acknowledge the discrepancy and consider the possibility of an error in the problem's options.
So, until next time, keep exploring the fascinating world of calculus, and don't be afraid to challenge yourself with tricky problems. You've got this!
Repair Input Keyword
What is the limit of the function (√(1 + x) - √(1 - x)) / x as x approaches 1? The possible answers are: A) 0, B) 1, C) 2, D) Infinity, E) Limit does not exist. Explain the steps to find the answer and which limit calculation techniques were used.
Title
Limit of (√(1 + x) - √(1 - x)) / x as x Approaches 1 A Calculus Problem
