Calculating Limits At Infinity A Step-by-Step Guide To Lim (x→∞) (x²+3x+2)/(x²+1)

Hey guys! Ever find yourself staring at a limit problem that seems intimidating? Let's break down one of those intimidating problems today. We are going to explore the limit of a rational function as x approaches infinity. Specifically, we're tackling the limit: lim (x→∞) (x²+3x+2)/(x²+1). This is a classic calculus problem that perfectly illustrates how to handle situations where both the numerator and denominator grow without bound. So, buckle up and let's dive in!

Understanding the Problem: What Does It Mean?

Before we jump into solving, let's make sure we understand what the problem is really asking. The expression lim (x→∞) (x²+3x+2)/(x²+1) is mathematical shorthand for asking, “What value does the fraction (x²+3x+2)/(x²+1) approach as x gets incredibly large?” Think of x as a slider that we're pushing further and further towards infinity. As x gets bigger, both the numerator (x²+3x+2) and the denominator (x²+1) also get bigger. But how do they grow relative to each other? Does the fraction as a whole approach a specific value, grow without bound, or perhaps oscillate? This is where the magic of limits comes in. Limits help us analyze the end behavior of functions, which is incredibly useful in many areas of mathematics, physics, and engineering. When we deal with infinity in limits, we're not treating infinity as a number, but rather as a concept representing unbounded growth. So, we are really interested in the trend of the function as x becomes arbitrarily large. Now, with the context set, let's investigate the techniques we can use to actually solve this limit.

The Divide-by-the-Highest-Power Technique: Our Key Strategy

Okay, so we know what the problem is asking, but how do we actually find the limit? The most effective strategy for dealing with limits of rational functions as x approaches infinity is the “divide by the highest power” technique. This technique leverages a clever algebraic manipulation to simplify the expression and make the limit more apparent. The core idea is to divide both the numerator and the denominator of the fraction by the highest power of x that appears in the entire expression. In our case, the highest power of x is x². So, we're going to divide every term in both the numerator and the denominator by x². Why does this work? Well, dividing by the highest power of x allows us to isolate the dominant terms in the expression. As x becomes very large, the terms with the highest powers will have the most significant impact on the function's behavior. By dividing through, we essentially “normalize” the expression, making it easier to see what happens as x approaches infinity. It's like comparing the sizes of two mountains by looking at their heights relative to sea level – dividing by the highest power provides a common reference point. This method transforms the original limit into a form where we can directly apply the properties of limits, particularly the rule that the limit of a constant divided by a power of x approaches zero as x approaches infinity. Now that we understand the strategy, let's apply it to our specific problem.

Step-by-Step Solution: Applying the Technique

Let’s put our strategy into action and solve the limit step by step. Remember, we’re working with: lim (x→∞) (x²+3x+2)/(x²+1). Our first move, as we discussed, is to divide both the numerator and the denominator by x². This gives us: lim (x→∞) [(x²/x²) + (3x/x²) + (2/x²)] / [(x²/x²) + (1/x²)]. Now, let's simplify each term: lim (x→∞) [1 + (3/x) + (2/x²)] / [1 + (1/x²)]. Notice what’s happening here. We’ve transformed the original fraction into a form where we have constants and terms that involve x in the denominator. This is exactly what we wanted! Now we can use the property of limits that says the limit of a quotient is the quotient of the limits (provided the limits exist and the denominator's limit isn't zero). We can also use the fact that the limit of a sum is the sum of the limits. So, let’s break it down further: [lim (x→∞) 1 + lim (x→∞) (3/x) + lim (x→∞) (2/x²)] / [lim (x→∞) 1 + lim (x→∞) (1/x²)]. Now comes the crucial part. As x approaches infinity, any constant divided by a power of x approaches zero. That is, lim (x→∞) (3/x) = 0, lim (x→∞) (2/x²) = 0, and lim (x→∞) (1/x²) = 0. Substituting these limits, we get: [1 + 0 + 0] / [1 + 0] = 1/1 = 1. Therefore, the limit of the given rational function as x approaches infinity is 1. Isn't it satisfying how a seemingly complex problem simplifies down to a neat solution with the right technique?

Visualizing the Result: What Does the Graph Tell Us?

It's always a good idea to connect the mathematical solution with a visual representation. So, let's think about what the graph of the function f(x) = (x²+3x+2)/(x²+1) looks like. Visualizing the graph can give us an intuitive understanding of why the limit is 1. If you were to graph this function (you can use a graphing calculator or online tool), you'd see that as x gets larger and larger (either positively or negatively), the graph of the function gets closer and closer to the horizontal line y = 1. This horizontal line is called a horizontal asymptote. A horizontal asymptote represents the value that a function approaches as x approaches positive or negative infinity. In our case, the horizontal asymptote is y = 1, which confirms our calculated limit. The graph visually demonstrates that the ratio of the numerator and denominator is approaching 1 as x becomes very large. The 3x and +2 in the numerator, and the +1 in the denominator, become insignificant compared to the x² terms when x is huge. This graphical perspective reinforces our algebraic solution and helps build a stronger understanding of the concept of limits at infinity.

Why This Matters: Real-World Applications

Okay, so we've solved the limit problem, but you might be wondering,