Calculate Limit Of (3n+2)/(2n+7) As N→∞

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Hey guys! Today, let's dive into a classic problem from calculus: finding the limit of a sequence. Specifically, we're going to figure out what happens to the expression (3n + 2) / (2n + 7) as n gets incredibly large – approaching infinity. This type of problem is fundamental in understanding how functions and sequences behave as their inputs grow without bound. So, grab your thinking caps, and let’s get started!

Understanding Limits

Before we jump into the nitty-gritty, let's quickly recap what a limit actually means. In simple terms, the limit of a sequence or function tells us the value that the sequence or function "approaches" as the input (in our case, n) gets closer and closer to a specific value (in our case, infinity). It's not necessarily the actual value at infinity (since infinity isn't a number), but rather the value it's heading towards. Understanding limits is crucial because it forms the bedrock of calculus, allowing us to discuss concepts like continuity, derivatives, and integrals rigorously.

When we talk about limits approaching infinity, we're interested in the end behavior of the function. What does the function do as n becomes a massive number? Does it increase without bound, decrease towards zero, or settle down to a specific value? Finding these limits often involves algebraic manipulation and a bit of intuition. In our problem, we want to evaluate the limit as n tends to infinity of the expression (3n + 2) / (2n + 7). This means we want to see what value this fraction approaches as n gets larger and larger. This type of problem frequently shows up in calculus courses and is a great way to practice your algebraic skills and your understanding of limits. The process usually involves identifying the dominant terms in the numerator and the denominator and then simplifying the expression to make the limit evaluation straightforward.

Step-by-Step Solution

Okay, let's tackle the problem head-on. Here’s how we can find the limit:

1. Identify the Dominant Terms

When n gets super big, the terms with the highest power of n dominate the expression. In our case, that's 3n in the numerator and 2n in the denominator. The constants (+2 and +7) become insignificant compared to these terms as n approaches infinity. Identifying these dominant terms is crucial, because it allows us to simplify the expression and focus on what really matters when n gets extremely large.

2. Divide by the Highest Power of n

To simplify things, we'll divide both the numerator and the denominator by the highest power of n present, which is n in this case. This is a standard technique when evaluating limits at infinity for rational functions (polynomials divided by polynomials). Dividing by the highest power allows us to isolate the constant coefficients of the dominant terms and eliminate the indeterminate form that we would otherwise encounter when directly substituting infinity. So, we divide both the numerator and the denominator by n:

lim (n→∞) [ (3n + 2) / n ] / [ (2n + 7) / n ]

3. Simplify the Expression

Now, let's simplify the expression by distributing the division:

lim (n→∞) (3 + 2/n) / (2 + 7/n)

Notice what happened? We've transformed the original expression into one where the terms 2/n and 7/n appear. As n approaches infinity, these terms will approach zero. This simplification is key to evaluating the limit.

4. Evaluate the Limit

As n approaches infinity, the terms 2/n and 7/n approach 0. So, we can rewrite the limit as:

lim (n→∞) (3 + 0) / (2 + 0) = 3/2

Therefore, the limit of the sequence as n approaches infinity is 3/2.

Why This Works: A Deeper Look

You might be wondering, why can we just ignore those constant terms and the 2/n and 7/n terms? The key is understanding the concept of relative size as n grows without bound. Think of it this way: imagine you have a million dollars, and then you find a five-dollar bill on the street. Does that five dollars significantly change your overall wealth? Not really. Similarly, as n gets incredibly large, adding a constant or dividing by n makes a negligible difference compared to the dominant terms.

More formally, we're using the limit properties that state the limit of a sum is the sum of the limits (provided those limits exist) and the limit of a constant times a function is the constant times the limit of the function. By dividing by the highest power of n, we're essentially isolating the parts of the expression that contribute most significantly to the limit as n approaches infinity. This technique is widely used in calculus and analysis for evaluating limits of rational functions and other types of expressions.

Alternative Methods

While the method described above is the most common and straightforward, there are alternative approaches you could use to solve this problem.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits of indeterminate forms, such as 0/0 or ∞/∞. While it's technically applicable here, it's a bit overkill for this particular problem. L'Hôpital's Rule states that if the limit of f(x)/g(x) as x approaches c is of the form 0/0 or ∞/∞, then the limit is equal to 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, we could apply L'Hôpital's Rule by taking the derivatives of the numerator and the denominator:

lim (n→∞) (3n + 2) / (2n + 7) = lim (n→∞) 3 / 2 = 3/2

As you can see, L'Hôpital's Rule gives us the same answer, but it involves taking derivatives, which is unnecessary for this simple problem. It's generally better to use L'Hôpital's Rule when you encounter more complex indeterminate forms that cannot be easily simplified algebraically.

Intuitive Approach

For this specific problem, you could also use a more intuitive approach. As n becomes extremely large, the +2 and +7 become insignificant. So, the expression (3n + 2) / (2n + 7) essentially becomes 3n / 2n, which simplifies to 3/2. This approach is less rigorous but can be helpful for quickly estimating the limit.

Common Pitfalls

When evaluating limits, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

  • Ignoring Dominant Terms Too Early: Make sure you're only ignoring terms that are truly insignificant compared to the dominant terms as n approaches infinity. Don't just arbitrarily drop terms without justification.
  • Incorrectly Applying L'Hôpital's Rule: L'Hôpital's Rule only applies to indeterminate forms of the type 0/0 or ∞/∞. Make sure you've verified that the limit is of this form before applying the rule.
  • Algebraic Errors: Be careful with your algebra! A simple mistake in simplifying the expression can lead to an incorrect answer.
  • Not Understanding the Concept of a Limit: Remember that a limit is the value a function approaches, not necessarily the value it equals at a particular point. This is especially important when dealing with limits at infinity.

Conclusion

So, there you have it! The limit of (3n + 2) / (2n + 7) as n approaches infinity is 3/2. We found this by identifying the dominant terms, dividing by the highest power of n, simplifying the expression, and then evaluating the limit. Remember, understanding limits is a fundamental concept in calculus, and mastering these techniques will help you tackle more complex problems in the future. Keep practicing, and you'll become a limit-calculating pro in no time! Hope this helped, and happy calculating!