Prove Sequence Limit: (5n+3)/(2n-4) = 5/2

Oct 16, 2025 by ADMIN 42 views

Proving the Limit of a Sequence: (5n+3)/(2n-4) = 5/2

Hey guys! Today, we're diving into the exciting world of sequence limits and proving one using the formal definition. We'll tackle the sequence (5n+3)/(2n-4) and demonstrate that its limit as n approaches infinity is indeed 5/2. This might sound a bit intimidating at first, but trust me, we'll break it down step by step so it's super clear. So, buckle up and let's get started!

Understanding the Epsilon-N Definition of a Limit

Before we jump into the proof itself, let's quickly refresh the fundamental concept we'll be using: the epsilon-N definition of a limit. This definition provides a rigorous way to express what it means for a sequence to converge to a specific value. In simpler terms, it formalizes the idea that the terms of the sequence get arbitrarily close to the limit as n becomes sufficiently large. This is the backbone of how we formally demonstrate convergence in mathematical analysis. The epsilon-N definition is what we use to rigorously define limits. A sequence a_n converges to a limit L if, for any positive number ε (epsilon), no matter how small, there exists a positive integer N such that for all n > N, the absolute difference between a_n and L is less than ε. Mathematically, this is expressed as:

| a_n - L | < ε for all n > N

Where:

ε (epsilon): A small positive number representing the desired level of closeness to the limit.
N: A positive integer indicating the point beyond which all terms of the sequence are within ε of the limit.

Think of ε as a tolerance level. We want to show that we can make the terms of the sequence as close to the limit as we want (within ε) by going far enough out in the sequence (beyond N). This 'arbitrarily close' aspect is crucial. It means no matter how tiny we make ε, we can always find an N that satisfies the condition. This definition essentially captures the intuitive idea of a limit: as n gets larger, the terms of the sequence get closer and closer to the limit L. We're not just saying they get close; we're saying we can force them to be arbitrarily close. This is why the epsilon-N definition is so powerful – it provides a precise and unambiguous way to talk about convergence. For a sequence to truly converge, it must satisfy this definition. There can't be any loopholes or exceptions. No matter how small an ε you choose, there must be an N that works. If you can't find such an N, the sequence doesn't converge to that limit. It might diverge, oscillate, or behave in some other way.

Problem Statement: lim (as n approaches ∞) of (5n+3)/(2n-4) = 5/2

Now, let's restate the problem we're tackling today. We need to prove, using the epsilon-N definition, that the limit of the sequence (5n+3)/(2n-4) as n approaches infinity is equal to 5/2. In mathematical notation, this is written as:

lim (n→∞) (5n+3)/(2n-4) = 5/2

To prove this, we need to show that for any ε > 0, there exists an integer N such that for all n > N, the absolute value of the difference between (5n+3)/(2n-4) and 5/2 is less than ε. In simpler terms, we want to demonstrate that we can make the sequence as close to 5/2 as we want by choosing a sufficiently large value of n. This involves a bit of algebraic manipulation and careful reasoning. We'll start by working with the expression |(5n+3)/(2n-4) - 5/2| and try to simplify it. The goal is to get it into a form where we can easily see how it relates to n and how we can make it less than ε by choosing a suitable N.

The Proof: A Step-by-Step Walkthrough

Okay, let's dive into the heart of the matter: the proof itself. We'll take it one step at a time to make sure everything is crystal clear.

Step 1: Setting up the Inequality

First, we need to start with the absolute value expression from the epsilon-N definition. We want to show that:

|(5n+3)/(2n-4) - 5/2| < ε for all n > N

This is the inequality we need to work with. Our goal is to manipulate the left-hand side until we can isolate n and figure out how large N needs to be.

Step 2: Simplifying the Expression

Let's simplify the expression inside the absolute value. We'll start by finding a common denominator:

|(5n+3)/(2n-4) - 5/2| = |(2(5n+3) - 5(2n-4)) / (2(2n-4))|

Now, let's expand the numerator:

| (10n + 6 - 10n + 20) / (4n - 8) |

Notice that the 10n terms cancel out, leaving us with:

| 26 / (4n - 8) |

We can simplify this further by factoring out a 2 from the denominator:

| 26 / (4(n - 2)) | = | 13 / (2(n - 2)) |

Since n approaches infinity, we can assume that n > 2, which means (n - 2) is positive. Therefore, we can drop the absolute value signs (but it's still good practice to keep them until we are sure the expression inside is positive):

13 / (2(n - 2))

This simplified expression is much easier to work with!

Step 3: Isolating n

Now, we want to find a condition on n that makes this expression less than ε. So, we set up the inequality:

13 / (2(n - 2)) < ε

To isolate n, we'll first multiply both sides by 2(n - 2) and divide both sides by ε (remembering that ε is positive, so we don't need to flip the inequality sign):

13 / ε < 2(n - 2)

Next, divide both sides by 2:

13 / (2ε) < n - 2

Finally, add 2 to both sides:

13 / (2ε) + 2 < n

So, we have found a condition on n: n must be greater than 13/(2ε) + 2.

Step 4: Choosing N

Remember the definition? We need to find an N such that for all n > N, the inequality holds. We've found that n > 13/(2ε) + 2. So, we can choose N to be any integer greater than or equal to 13/(2ε) + 2. A common and safe choice is to let N be the smallest integer greater than 13/(2ε) + 2. Mathematically, we can write this as:

N = ⌈13 / (2ε) + 2⌉

Where ⌈x⌉ represents the ceiling function, which gives the smallest integer greater than or equal to x. The choice of N is crucial. It dictates how far out in the sequence we need to go to ensure the terms are within ε of the limit. The formula N = ⌈13 / (2ε) + 2⌉ provides a concrete way to calculate N for any given ε. This formula is a direct result of our algebraic manipulation and ensures that our chosen N satisfies the epsilon-N definition.

Step 5: Formalizing the Proof

Now, let's put it all together in a formal proof:

Let ε > 0 be given. Choose N = ⌈13 / (2ε) + 2⌉. Then, for all n > N, we have:

n > 13 / (2ε) + 2

Subtracting 2 from both sides:

n - 2 > 13 / (2ε)

Multiplying both sides by 2 and taking the reciprocal (flipping the inequality sign because we are taking the reciprocal of positive numbers):

2 / (n - 2) < 2ε / 13

Multiplying both sides by 13/2:

13 / (2(n - 2)) < ε

Since 13 / (2(n - 2)) is equal to |(5n+3)/(2n-4) - 5/2|, we have:

|(5n+3)/(2n-4) - 5/2| < ε

Therefore, by the epsilon-N definition of a limit, lim (n→∞) (5n+3)/(2n-4) = 5/2.

Step 6: Concluding the proof.

Boom! We've officially proven that the limit of the sequence (5n+3)/(2n-4) as n approaches infinity is indeed 5/2, all thanks to the power of the epsilon-N definition. It might seem like a lot of steps, but each one is logical and builds upon the previous. With practice, these types of proofs become much more intuitive. This completes our journey through proving the limit of the sequence. We've successfully navigated the epsilon-N definition, performed the necessary algebraic manipulations, and arrived at the conclusion. The key takeaway is that by carefully choosing N based on ε, we can guarantee that the terms of the sequence get arbitrarily close to the limit as n gets large.

Key Takeaways

Let's recap the important things we've learned in this proof:

Epsilon-N Definition: We used the formal definition of a limit to rigorously prove the convergence of the sequence.
Algebraic Manipulation: Simplifying the expression inside the absolute value was crucial for isolating n.
Choosing N: The right choice of N is essential to satisfy the epsilon-N definition. We found a formula for N based on ε.
Formal Proof: We presented the proof in a clear and logical manner, demonstrating each step.

Remember, the epsilon-N definition might seem abstract at first, but it's a powerful tool for understanding and proving limits. Keep practicing, and you'll become a pro in no time! Understanding this concept is like unlocking a secret level in math. The epsilon-N definition is not just some abstract idea; it's a fundamental tool for working with limits and continuity in calculus and analysis. Mastering it opens doors to understanding more advanced concepts and tackling more challenging problems.

Additional Tips and Tricks

Here are a few extra tips to keep in mind when tackling epsilon-N proofs:

Start with the inequality: Always begin by setting up the inequality |a_n - L| < ε.
Simplify, simplify, simplify: Algebraic simplification is your best friend. The simpler the expression, the easier it is to isolate n.
Think backwards: Sometimes, it helps to work backwards from the desired inequality to figure out what N should be.
Don't be afraid to overestimate: It's okay to choose an N that's larger than necessary. The definition only requires that some N exists.
Practice makes perfect: The more proofs you do, the more comfortable you'll become with the process.

Conclusion

So there you have it! We've successfully proven that lim (as n approaches infinity) of (5n+3)/(2n-4) = 5/2 using the epsilon-N definition. I hope this step-by-step explanation has made the concept clearer for you guys. Remember, the key is to understand the definition, simplify the expression, and choose the right N. Keep practicing, and you'll be a limit-proving master in no time! Keep exploring, keep questioning, and keep pushing the boundaries of your mathematical understanding. Math is not just about formulas and equations; it's about thinking critically, solving problems, and making connections. Embrace the challenge, and you'll be amazed at what you can achieve!