Evaluating Limits At Infinity A Step-by-Step Guide

This article provides a comprehensive guide on how to evaluate the limit of the rational function $\lim _{x \rightarrow \infty} \frac{(2-x)(9+3 x)}{(3-3 x)(4+6 x)}$. We will explore the step-by-step process, including algebraic manipulation and simplification, to arrive at the final answer. This topic is a fundamental concept in calculus and is essential for understanding the behavior of functions as their input values grow infinitely large.

Introduction to Limits at Infinity

In calculus, limits at infinity are used to describe the behavior of a function as the input variable, in this case, x, approaches either positive or negative infinity. Understanding these limits is crucial in various fields such as physics, engineering, and economics, where we often need to analyze the long-term behavior of systems and models. Evaluating limits at infinity for rational functions involves comparing the degrees of the polynomials in the numerator and the denominator. The degree of a polynomial is the highest power of the variable in the polynomial.

When dealing with rational functions, which are ratios of two polynomials, the limit as x approaches infinity can take one of three forms:

1. If the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, the limit is 0.
2. If the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator, the limit is the ratio of the leading coefficients.
3. If the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, the limit is either positive or negative infinity (depending on the sign of the leading coefficients and the direction in which x is approaching infinity).

Step-by-Step Evaluation

To evaluate the given limit, $\lim _{x \rightarrow \infty} \frac{(2-x)(9+3 x)}{(3-3 x)(4+6 x)}$, we will follow these steps:

Step 1: Expand the Numerator and Denominator

First, we expand both the numerator and the denominator by multiplying the binomials:

Numerator:

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

Denominator:

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

Step 2: Rewrite the Rational Function

Now we rewrite the original limit with the expanded polynomials:

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

Step 3: Divide by the Highest Power of x

To evaluate the limit as x approaches infinity, we divide both the numerator and the denominator by the highest power of x that appears in the expression. In this case, the highest power is $x^2$:

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

Step 4: Simplify the Expression

Next, we simplify the fractions:

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

Step 5: Evaluate the Limit

As x approaches infinity, the terms $\frac{3}{x}$, $\frac{18}{x^2}$, $\frac{6}{x}$, and $\frac{12}{x^2}$ all approach 0. Therefore, the limit becomes:

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

Step 6: Final Simplification

Finally, we simplify the fraction:

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

Thus, the limit of the given rational function as x approaches infinity is $\frac{1}{6}$.

Alternative Method: Leading Coefficients

An alternative and quicker method to solve this limit problem is by considering the leading coefficients. When x approaches infinity, the terms with the highest powers of x dominate the behavior of the polynomials. Therefore, we can focus on these terms.

From the expanded form, we have:

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

We only need to consider the terms with $x^2$:

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

Now, we can simplify by canceling out $x^2$:

$\lim _{x \rightarrow \infty} \frac{-3}{-18} = \frac{1}{6}$

This method provides the same result more efficiently.

Common Mistakes and How to Avoid Them

When evaluating limits at infinity, several common mistakes can lead to incorrect results. Here are some of these mistakes and how to avoid them:

Mistake 1: Ignoring Lower-Order Terms

A frequent error is not focusing on the highest-degree terms when x approaches infinity. While it's true that these terms dominate the behavior of the function, neglecting them entirely in intermediate steps can lead to a wrong answer. Always expand and identify the highest-degree terms before simplifying.

How to Avoid: Ensure you correctly identify the highest power of x in both the numerator and the denominator before attempting to simplify the expression.

Mistake 2: Incorrectly Expanding Polynomials

Algebraic errors in expanding the numerator and the denominator can lead to a completely different expression. It is crucial to carefully multiply out the terms and double-check your work.

How to Avoid: Take your time when expanding polynomials. Use the distributive property meticulously and verify each term.

Mistake 3: Not Dividing by the Highest Power of x

Dividing both the numerator and the denominator by the highest power of x is a critical step in evaluating limits at infinity. Failing to do so, or doing it incorrectly, will prevent the lower-order terms from approaching zero, which is necessary for the simplification process.

How to Avoid: Always divide both the numerator and the denominator by the highest power of x present in the expression. This ensures that you can correctly evaluate the limit as x approaches infinity.

Mistake 4: Misunderstanding the Behavior of Terms as x Approaches Infinity

It’s essential to understand that as x approaches infinity, terms of the form $\frac{c}{x^n}$, where c is a constant and n is a positive integer, approach zero. Confusing this behavior can lead to incorrect simplifications.

How to Avoid: Remember the fundamental rule that as x becomes infinitely large, fractions with x in the denominator tend towards zero. This understanding is crucial for correctly evaluating limits at infinity.

Mistake 5: Incorrectly Applying Limit Laws

Limit laws are tools that help simplify the evaluation of limits, but they must be applied correctly. For example, the limit of a quotient is the quotient of the limits only if the limits in the denominator are not zero. Misapplying these laws can lead to incorrect results.

How to Avoid: Familiarize yourself with the limit laws and understand their conditions for applicability. Ensure that you are applying them correctly, especially when dealing with indeterminate forms.

Real-World Applications

Limits at infinity are not just theoretical concepts; they have numerous practical applications in various fields. Here are a few examples:

Physics

In physics, limits at infinity are used to analyze the long-term behavior of physical systems. For example, in projectile motion, we can use limits to determine the maximum distance a projectile can travel if air resistance is negligible. Similarly, in thermodynamics, limits can help us understand the equilibrium state of a system as time approaches infinity.

Engineering

Engineers use limits at infinity to design and analyze control systems. For instance, in designing a feedback control system, engineers need to ensure that the system remains stable as time goes to infinity. Limits help in determining the stability of such systems by examining their behavior over an infinite time horizon.

Economics

Economists use limits at infinity to model long-term economic trends. For example, they might use limits to predict the long-term growth rate of an economy or the sustainability of a financial model. Understanding how economic variables behave in the long run is crucial for policy-making and investment decisions.

Computer Science

In computer science, limits at infinity are used in algorithm analysis. The efficiency of an algorithm is often described in terms of its time complexity, which is a function of the input size. Limits help in understanding how the algorithm performs as the input size grows infinitely large, allowing for the comparison of different algorithms.

Environmental Science

Environmental scientists use limits at infinity to model the long-term effects of pollution and climate change. For example, they might use limits to predict how pollutant concentrations will change over time or to model the long-term impact of greenhouse gas emissions on global temperatures.

Conclusion

In this article, we have thoroughly evaluated the limit of the rational function $\lim _{x \rightarrow \infty} \frac{(2-x)(9+3 x)}{(3-3 x)(4+6 x)}$. We explored the step-by-step process, including expanding the polynomials, dividing by the highest power of x, and simplifying the expression. We also discussed an alternative method using leading coefficients, which offers a quicker way to arrive at the solution. Understanding limits at infinity is crucial not only in calculus but also in various real-world applications across diverse fields. By mastering these techniques, you can confidently tackle similar problems and apply them in practical scenarios.

Additionally, we highlighted common mistakes that students often make when evaluating limits at infinity and provided strategies to avoid them. These include incorrectly expanding polynomials, neglecting lower-order terms, not dividing by the highest power of x, misunderstanding the behavior of terms as x approaches infinity, and misapplying limit laws. By being mindful of these pitfalls, you can improve your accuracy and problem-solving skills.

By following the methods and guidelines outlined in this article, you can enhance your understanding of limits at infinity and their applications. Remember to practice regularly and apply these concepts to various problems to reinforce your learning.