Problem 16 - Precalculus: Mathematics for Calculus

Question

Find the limit. $$\lim _{x \rightarrow-\infty}\left(\frac{3-x}{3+x}-2\right)$$

Step-by-Step Solution

Answer

The limit is -3.

Step 1: Simplify the expression

Start by rewriting the expression inside the limit, \( \frac{3-x}{3+x} - 2 \). We will combine the terms into a single fraction. The expression becomes: \[ \frac{3-x - 2(3+x)}{3+x} \] After simplifying the numerator, we get: \[ \frac{3-x - 6 - 2x}{3+x} = \frac{-1 - 3x}{3+x} \] The expression inside the limit now is \( \lim_{x \to -\infty} \frac{-1 - 3x}{3+x} \).

Step 2: Factor out the highest power of x

Notice that both numerator and the denominator have expressions that depend on \(x\). Factor out the highest power of \(x\), which is \(x\), from both to simplify the evaluation of the limit: \[ \frac{-1 - 3x}{3+x} = \frac{x(-\frac{1}{x} - 3)}{x(\frac{3}{x}+1)} \] Simplifying gives: \[ \frac{-\frac{1}{x} - 3}{\frac{3}{x} + 1} \]

Step 3: Evaluate the limit as x approaches negative infinity

As \(x\to -\infty\), the terms \(-\frac{1}{x}\) and \(\frac{3}{x}\) approach 0 because they are fractions with larger denominators as \(x\) becomes larger in magnitude. Therefore the expression simplifies to: \[ \lim_{x \to -\infty} \frac{-3}{1} = -3 \]

Step 4: Conclude the result

The limit of the given expression as \(x\) approaches negative infinity is \(-3\). This means that as \(x\) becomes larger in magnitude in the negative direction, the expression approaches -3.

Key Concepts

Negative Infinity

The concept of negative infinity is an essential aspect of limits and calculus. When we discuss limits as a variable approaches negative infinity, we refer to the behavior of a function as the input grows increasingly large in the negative direction. This can help us understand long-term trends or the asymptotic behavior of functions.In practical terms, if we're assessing the limit \( \lim_{x \to -\infty} f(x) \), we're asking what value \( f(x) \) gets close to as \( x \) becomes very large and negative. It is crucial to note that negative infinity \(-\infty\) is not a specific value but a concept representing that a variable is decreasing without bound. Grasping this idea helps in understanding how functions behave under extreme conditions.

Fraction Simplification

Fraction simplification is a useful technique in mathematics, especially when dealing with limits and algebraic expressions. When simplifying fractions, the goal is to rewrite the fraction in its simplest form. This involves combining or reducing terms to make the expression easier to analyze or solve.In the context of limits, simplifying fractions allows us to more clearly see how the numerator and the denominator behave as the variable approaches a certain value. In our exercise, the expression \( \frac{3-x}{3+x} - 2 \) is transformed into a single simplified fraction:

• Combine the fractions into: \( \frac{-1 - 3x}{3 + x} \).
• This simplification clarifies the effect of each term as \( x \) approaches negative infinity, facilitating the limit calculation.

Fraction simplification plays an important role when attempting to find the limit of complex expressions by making the underlying calculations and transformations more manageable.

Highest Power of x

Recognizing and factoring out the highest power of \( x \) is a crucial technique for evaluating limits, especially when \( x \) approaches infinity or negative infinity. By identifying the dominant term in both the numerator and denominator, we can simplify the expression significantly.In the provided exercise, both the numerator \(-1 - 3x\) and the denominator \(3 + x\) contain terms involving \( x \). The highest power of \( x \) is 1, as \( x \) is raised to the first power. By factoring \( x \) out, the expression becomes:\[ \frac{x(-\frac{1}{x} - 3)}{x(\frac{3}{x}+1)} \]This step simplifies the limit analysis because we can cancel \( x \) from the numerator and denominator. Thus, the problem reduces to evaluating a simpler fraction, which makes the limit computation more straightforward. Identifying and factoring the highest power of \( x \) helps in isolating dominant terms, making the limit easier to determine.

Precalculus

Precalculus serves as the foundation for calculus, involving concepts such as algebra, trigonometry, and the beginnings of limits and continuity. It prepares students for more advanced mathematical analysis featured in calculus courses.In precalculus, students learn how to manipulate and understand functions, build skills in algebraic simplification, and develop the ability to understand limits—a key component of calculus.One of the main goals of precalculus is to familiarize students with foundational skills needed for calculus. In the context of the given problem, understanding how to simplify expressions and apply the concept of limits as \( x \) approaches negative infinity exemplifies key precalculus techniques. These techniques help transition smoothly into solving more complex calculus problems, reinforcing skills such as function behavior analysis and fraction manipulation. Precalculus provides the essential toolkit needed for students to tackle calculus problems effectively.