Problem 8 - Precalculus Mathematics for Calculus

Question

Find the limit. $$\lim _{x \rightarrow \infty} \frac{2-3 x}{4 x+5}$$

Step-by-Step Solution

Answer

The limit is \( \frac{-3}{4} \).

Step 1: Identify Variables and Expression

The expression given is \( \frac{2 - 3x}{4x + 5} \), and we need to find the limit as \( x \) approaches infinity.

Step 2: Determine Terms Dominating the Expression

As \( x \to \infty \), the degree of the terms \(-3x\) in the numerator and \(4x\) in the denominator will dominate because they are the highest degree terms.

Step 3: Simplify the Expression

Divide each term in both the numerator and the denominator by \( x \), the term with the highest degree inside the expression:\[\frac{2 - 3x}{4x + 5} = \frac{\frac{2}{x} - 3}{4 + \frac{5}{x}}\].

Step 4: Take the Limit as x Approaches Infinity

As \( x \rightarrow \infty \), the terms \( \frac{2}{x} \) and \( \frac{5}{x} \) approach 0. Substitute these into the expression from Step 3:\[\lim_{x \to \infty} \frac{\frac{2}{x} - 3}{4 + \frac{5}{x}} = \frac{0 - 3}{4 + 0} = \frac{-3}{4}.\]

Step 5: State the Final Answer

The limit of the expression as \( x \to \infty \) is \( \frac{-3}{4} \).

Key Concepts

Asymptotic Behavior

Asymptotic behavior in mathematics refers to the behavior of functions as input values become very large, or in some cases exceedingly small. This is especially important in calculus when evaluating limits, like the one from our example, where the limit of a rational function is taken as the variable approaches infinity.
Understanding asymptotic behavior helps us figure out what happens to our function as it moves towards these extreme values. Rather than examining the entire function, we focus on the dominant terms, or those that most influence the function's behavior for large inputs. This is because the non-dominant terms typically tend toward zero and become insignificant in the overall function. 

• In the example, as \(x\) gets very large, the terms \(\frac{2}{x}\) and \(\frac{5}{x}\) become negligible.
• The function simplifies by focusing only on the dominant terms \(-3x\) and \(4x\).

By understanding asymptotic behavior, we can simplify and accurately determine the limit of functions as they approach boundaries or infinity.

Rational Functions

Rational functions are algebraic expressions represented as the ratio of two polynomials. Understanding these functions is crucial because they are frequently used in calculus for solving limits and finding asymptotes.
For a rational function in the form \(\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials, the behavior of the function as it approaches infinity often depends on the degrees of these polynomials.

• If the degree of \(p(x)\) is less than the degree of \(q(x)\), the function approaches zero.
• If both have the same degree, the function approaches the ratio of their leading coefficients.
• If the degree of \(p(x)\) is greater, the function tends towards infinity.

In our example, both the numerator and denominator have linear terms \(-3x\) and \(4x\). As such, the limit at infinity is determined by the leading coefficients, resulting in \(\frac{-3}{4}\). Recognizing these key aspects of rational functions is essential in calculus for evaluating limits and understanding function behavior.

Infinity in Calculus

The concept of infinity is integral in calculus, especially when evaluating limits. Infinity signifies an unbounded quantity, or a value that continues indefinitely without converging on a specific point.
When dealing with limits, we often encounter infinity as a boundary. It's important to understand how to handle it properly. Calculus provides the tools to reason about what happens to functions as they explore these infinite domains.

• In our problem, we are interested in the behavior of the function as \(x\) increases without bound.
• Infinity isn't a number, but rather a way to talk about limits that describe unending growth or decrease.

In practical terms, as functions approach infinity, we simplify them by focusing on terms with the highest impact while ignoring those that diminish. This allows us to understand and compute the limits in a more approachable manner.