Evaluating Limits At Infinity A Step-by-Step Guide

$$\lim _{x \rightarrow \infty} \frac{(2-x)(9+3 x)}{(3-3 x)(4+6 x)}$$

In the realm of calculus, evaluating limits is a fundamental skill. This article delves into the process of evaluating the limit of a rational function as $x$ approaches infinity. The specific function we will analyze is:

$$\lim _{x \rightarrow \infty} \frac{(2-x)(9+3 x)}{(3-3 x)(4+6 x)}$$

This problem falls under the category of limits at infinity, a crucial topic in calculus that helps us understand the behavior of functions as their input grows without bound. Mastering these techniques is essential for understanding concepts like asymptotes and the end behavior of functions.

Understanding Limits at Infinity

Before we dive into the solution, let's establish a clear understanding of what limits at infinity represent. When we say $\lim_{x \rightarrow \infty} f(x) = L$, we are essentially stating that as $x$ becomes arbitrarily large, the function $f(x)$ gets closer and closer to the value $L$. This value $L$ can be a finite number, infinity, or even negative infinity. In the context of rational functions, which are ratios of polynomials, limits at infinity often reveal the function's long-term behavior and any horizontal asymptotes it might possess.

To effectively evaluate limits at infinity for rational functions, we employ a strategy focused on identifying the dominant terms. The dominant terms are those with the highest powers of $x$ in the numerator and the denominator. These terms dictate the function's behavior as $x$ grows extremely large because they outgrow the influence of lower-order terms. For instance, in a polynomial like $3x^2 + 2x + 1$, the term $3x^2$ is dominant as $x$ approaches infinity.

The general approach involves dividing both the numerator and the denominator by the highest power of $x$ present in the denominator. This manipulation simplifies the expression and allows us to analyze the limit as $x$ approaches infinity more easily. Terms with lower powers of $x$ in the numerator will approach zero, leaving us with a simplified expression involving the dominant terms. This technique allows us to effectively "zoom out" and see the function's overall trend as $x$ becomes very large.

Step-by-Step Solution

Now, let's apply this strategy to our given limit:

$$\lim _{x \rightarrow \infty} \frac{(2-x)(9+3 x)}{(3-3 x)(4+6 x)}$$

Step 1: Expand the Numerator and Denominator

First, we need to expand both the numerator and the denominator to identify the terms with the highest powers of $x$:

Numerator:

$$(2-x)(9+3x) = 2(9) + 2(3x) - x(9) - x(3x) = 18 + 6x - 9x - 3x^2 = -3x^2 - 3x + 18$$

Denominator:

$$(3-3x)(4+6x) = 3(4) + 3(6x) - 3x(4) - 3x(6x) = 12 + 18x - 12x - 18x^2 = -18x^2 + 6x + 12$$

So, our limit now looks like this:

$$\lim _{x \rightarrow \infty} \frac{-3x^2 - 3x + 18}{-18x^2 + 6x + 12}$$

Step 2: Identify the Highest Power of x

In both the numerator and the denominator, the highest power of $x$ is $x^2$. This identifies the dominant terms that will influence the limit as $x$ approaches infinity.

Step 3: Divide by the Highest Power of x

Next, we divide both the numerator and the denominator by $x^2$:

$$\lim _{x \rightarrow \infty} \frac{\frac{-3x^2}{x^2} - \frac{3x}{x^2} + \frac{18}{x^2}}{\frac{-18x^2}{x^2} + \frac{6x}{x^2} + \frac{12}{x^2}}$$

Simplifying, we get:

$$\lim _{x \rightarrow \infty} \frac{-3 - \frac{3}{x} + \frac{18}{x^2}}{-18 + \frac{6}{x} + \frac{12}{x^2}}$$

Step 4: Evaluate the Limit as x Approaches Infinity

As $x$ approaches infinity, the terms $\frac{3}{x}$, $\frac{18}{x^2}$, $\frac{6}{x}$, and $\frac{12}{x^2}$ all approach zero. This is because dividing a constant by an increasingly large number results in a value that gets closer and closer to zero. Therefore, we can rewrite the limit as:

$$\lim _{x \rightarrow \infty} \frac{-3 - 0 + 0}{-18 + 0 + 0} = \frac{-3}{-18}$$

Step 5: Simplify the Result

Finally, we simplify the fraction:

$$\frac{-3}{-18} = \frac{1}{6}$$

Therefore, the limit is:

$$\lim _{x \rightarrow \infty} \frac{(2-x)(9+3 x)}{(3-3 x)(4+6 x)} = \frac{1}{6}$$

Conclusion

In this article, we successfully evaluated the limit of the given rational function as $x$ approaches infinity. The key steps involved expanding the numerator and denominator, identifying the highest power of $x$, dividing by that power, and then evaluating the limit as $x$ approaches infinity. This process highlights the importance of understanding dominant terms in determining the long-term behavior of rational functions. The result, $\frac{1}{6}$, signifies that as $x$ becomes infinitely large, the function approaches this value, which can also be interpreted as a horizontal asymptote of the function's graph.

Understanding limits at infinity is not just an exercise in calculus; it's a fundamental tool for analyzing the behavior of functions and modeling real-world phenomena. From physics to economics, understanding how functions behave at extreme values is crucial for making predictions and informed decisions. This skill is essential for anyone pursuing advanced studies in mathematics, science, or engineering.

The technique of dividing by the highest power of $x$ is a powerful method that can be applied to a wide range of rational functions. It allows us to effectively isolate the dominant terms and determine the limit as $x$ approaches infinity. This method provides a systematic way to analyze the end behavior of functions and identify horizontal asymptotes.

By mastering these techniques, you'll gain a deeper understanding of how functions behave and how to apply calculus to solve real-world problems. The ability to confidently evaluate limits at infinity is a valuable asset in any mathematical or scientific endeavor. This skill is essential for understanding concepts like convergence and divergence, which are fundamental in areas such as series and sequences, differential equations, and mathematical modeling. The limit we evaluated, $\frac{1}{6}$, represents a specific value that the function approaches as $x$ grows without bound. This type of analysis is critical in various applications, such as determining the stability of systems, the long-term behavior of populations, and the efficiency of algorithms.

• Evaluate the limit
• Limits at infinity
• Rational functions
• Dominant terms
• Highest power of x
• Horizontal asymptotes
• End behavior of functions
• Calculus techniques
• Step-by-step solution
• Mathematical analysis