Problem 4 - Calculus for Biology and Medicine

Question

Evaluate the limits. $$ \lim _{x \rightarrow-\infty} \frac{2 x+1}{3-4 x} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{2}\).

Step 1: Identify the Limit Type

The limit to evaluate is \( \lim _{x \rightarrow -\infty} \frac{2x+1}{3-4x} \). Since \( x \to -\infty \) indicates that \( x \) is approaching negative infinity, we are considering the behavior of a rational function as \( x \) becomes very large in the negative direction.

Step 2: Analyze Degrees of Polynomials

In the rational function \( \frac{2x+1}{3-4x} \), the numerator has a degree of 1 (\( 2x+1 \)) and the denominator also has a degree of 1 (\( 3-4x \)). Since both the numerator and the denominator have the same degree, the limit as \( x \to -\infty \) will be determined by the leading coefficients.

Step 3: Determine the Leading Coefficients

The leading coefficient of the numerator \( 2x \) is 2, and the leading coefficient of the denominator \( -4x \) is -4. Since the degrees are the same, the limit will equal the ratio of the leading coefficients. Therefore, the limit is \( \frac{2}{-4} \).

Step 4: Simplify the Ratio

Simplify the ratio \( \frac{2}{-4} \) to find the limit. The simplification yields \( \frac{2}{-4} = -\frac{1}{2} \).

Step 5: State the Final Result

The evaluated limit as \( x \to -\infty \) for the given function is \(-\frac{1}{2}\). Therefore, \( \lim _{x \rightarrow -\infty} \frac{2x+1}{3-4x} = -\frac{1}{2} \).

Key Concepts

Rational Function

A rational function is a type of fraction where both the numerator and denominator are polynomials. Polynomials are mathematical expressions that involve variables raised to various powers, each multiplied by coefficients. The general form of a rational function is \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial functions and \( Q(x) eq 0 \).
Rational functions appear often in mathematics due to their versatile nature. They are predominately used in calculus to explore limits and continuity. In our exercise, the function \( \frac{2x+1}{3-4x} \) is a rational function because it is a ratio of two polynomials: \( 2x+1 \) and \( 3-4x \).
Rational functions have interesting properties, especially regarding limits. As \( x \) approaches infinity or negative infinity, their behavior is largely determined by the degrees and leading coefficients of the numerator and denominator polynomials.

Leading Coefficients

In a polynomial, the leading coefficient is the number in front of the variable with the highest exponent. This coefficient is crucial for determining the end behavior of a polynomial function and, by extension, a rational function.
For our exercise, the rational function is \( \frac{2x+1}{3-4x} \):

• The leading coefficient of the numerator \(2x+1\) is 2.
• The leading coefficient of the denominator \(3-4x\) is -4.

When you evaluate limits for rational functions as \( x \) approaches infinity or negative infinity, and both the numerator and the denominator have the same degree, the limit is simply the ratio of these leading coefficients. This makes the analysis straightforward, as shown in the original exercise where this ratio, \( \frac{2}{-4} \), simplifies to \( -\frac{1}{2} \). This understanding allows you to compute limits swiftly once you identify the leading coefficients.

Polynomial Degree Analysis

In polynomial degree analysis, the degree of a polynomial is the highest power of the variable present in the polynomial. This concept is essential when dealing with rational functions, especially in the context of limits at infinity.
For the given rational function \( \frac{2x+1}{3-4x} \), we analyze the degrees:

• The polynomial \( 2x+1 \) has a degree of 1 because the highest power of \( x \) is 1.
• Similarly, the polynomial \( 3-4x \) also has a degree of 1.

When both the numerator and the denominator have the same degree, the limit at infinity is determined by the leading coefficients of the terms with the highest degree. This is because, as \( x \) grows large, the lower degree terms become negligible, and the behavior of the function mimics the ratio of these leading coefficients. This insight simplifies the process of evaluating limits, especially in cases where such a straightforward comparison is possible.