Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Write an expression for a function \(f(x)\) with the given features. \(f(x)\) is a quotient of two polynomials of degree greater than \(2, \lim _{x \rightarrow \infty} f(x)=0\)

Step-by-Step Solution

Answer

Example function: \( f(x) = \frac{x^3 + 2x^2 + 3x + 4}{x^4 + 5x^3 + 6x^2 + 7x + 8} \).

Step 1: Choosing the Function Type

Since we need a quotient of two polynomials (a rational function), we can represent it as a fraction \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.

Step 2: Degree of Polynomials

To satisfy the condition that each polynomial should be of degree greater than 2, we choose, for instance, \( P(x) = a_n x^n + \, ... \, + a_3 x^3 \) and \( Q(x) = b_m x^m + \, ... \, + b_3 x^3 \), with \( n, m > 2 \).

Step 3: Limit at Infinity Condition

The condition \( \lim_{x \to \infty} f(x) = 0 \) implies that the degree of the polynomial in the denominator must be greater than the degree of the polynomial in the numerator, i.e., \( m > n \).

Step 4: Formulating the Function

Assuming \( n = 3 \) and \( m = 4 \), we can create a simple function such as \( f(x) = \frac{x^3 + 2x^2 + 3x + 4}{x^4 + 5x^3 + 6x^2 + 7x + 8} \). This satisfies \( n = 3, m = 4 \) with \( m > n \).

Step 5: Verification

To verify, check the degrees of the numerator and denominator: numerator is 3 and denominator is 4. Therefore, \( \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{x^3}{x^4} = \lim_{x \to \infty} \frac{1}{x} = 0 \).

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, involving terms in the form of multitudes of powers. They can have one term (monomials), two terms (binomials), or more. A polynomial in expressed form like \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\), where each \( a_i \) is a coefficient and \( n \) is a non-negative integer representing the degree. These expressions are fundamental in algebra and calculus, serving as building blocks for more complex equations.Key characteristics of polynomials include:

• The coefficients can be whole numbers, fractions, or decimals.
• The variables in these expressions are raised to whole number exponents.
• They generally represent smooth, continuous functions.

Understanding polynomials is crucial for many branches of mathematics as they help in solving algebraic equations, modeling real-world phenomena, and even in optimizing functions.

Degree of Polynomials

The degree of a polynomial is the highest power of the variable \( x \) that has a non-zero coefficient. This degree provides valuable information about the behavior and characteristics of the polynomial, such as its growth rate and shape.Consider the polynomial \(P(x) = 3x^4 + 2x^2 + x + 7\). Here, the highest power of \( x \) is 4, so the degree of this polynomial is 4. This implies that it is a quartic polynomial.Knowing the degree allows you to:

• Predict the number of roots or solutions the polynomial can have — a polynomial of degree \( n \) can have up to \( n \) roots.
• Determine the basic shape of the polynomial graph, such as whether it opens upwards or downwards.
• Assess the polynomial's end behavior which describes how the function behaves as \( x \) approaches positive or negative infinity.

In context of rational functions, comparing the degrees of the numerator and denominator helps us understand their limits as we approach infinity.

Limits at Infinity

Limits at infinity explore the behavior of functions as the variable \( x \) increases or decreases without bound. For rational functions, which are the quotient of two polynomials, these limits help determine how the function behaves far from the origin.Consider a rational function \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials. The degree of the numerator and denominator is crucial in finding the limit at infinity:

• If the degree of \(P(x)\) is less than the degree of \(Q(x)\), the limit as \(x\) approaches infinity is zero: \(\lim_{x \to \infty} f(x) = 0\).
• If the degree of \(P(x)\) is equal to the degree of \(Q(x)\), the limit is the ratio of their leading coefficients.
• If the degree of \(P(x)\) is greater than the degree of \(Q(x)\), the limit does not exist in regular terms as it approaches infinity or negative infinity.

By understanding these relationships, we gain insights into the function's end behavior, which aids in graphing and contextualizing mathematical models.