Problem 62 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 59-62, determine whether the statement is true or false. Justify your answer. When the degrees of the numerator and denominator of a rational function are equal, the limit does not exist.

Step-by-Step Solution

Answer

The statement is false. When the degrees of the numerator and denominator of a rational function are equal, the limit exists and is simply the ratio of the leading coefficients of the numerator and the denominator.

Step 1: Identify the Statement Claim

The statement that needs to be evaluated for truthfulness is: 'When the degrees of the numerator and denominator of a rational function are equal, the limit does not exist.'

Step 2: Understand the Rational Function

A rational function is a fraction of two polynomial functions. Because the degrees of the numerator and denominator are stated to be equal, this rational function is of the form \( \frac{a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0}{b_nx^n + b_{n-1}x^{n-1} + ... + b_1x + b_0} \), where \( n \) is the degree of both the numerator and the denominator.

Step 3: Analyze Limit of Rational Function

Now let's analyze the limit as \( x \) approaches infinity. By dividing every term in the numerator and the denominator by \( x^n \), we obtain \( \frac{a_n + a_{n-1}/x + ... + a_1/x^{n-1} + a_0/x^n}{b_n + b_{n-1}/x + ... + b_1/x^{n-1} + b_0/x^n} \). As \( x \) goes to infinity, all the terms divided by \( x \) will go to 0, so the limit is \( a_n/b_n \). The same applies when \( x \) approaches negative infinity. That means, the limits exist and are equal to the ratio of the leading coefficients.

Step 4: Evaluate the Statement

Considering the analysis stated above, it is clear that the initial statement given in the exercise, 'When the degrees of the numerator and denominator of a rational function are equal, the limit does not exist', is false. A limit does exist and it is equal to the ratio of the leading coefficients.

Key Concepts

Limits at Infinity

When we talk about limits at infinity, we want to understand what happens to a function as the input value, usually represented by \(x\), goes beyond all bounds—both positively and negatively. In terms of rational functions, which are ratios of polynomial functions, it's crucial to analyze how they behave as \(x\) grows very large or very small.

For a rational function of the form \( R(x) = \frac{P(x)}{Q(x)} \), determining the limit as \(x\) approaches infinity depends on the degrees of the polynomials in the numerator and the denominator:

• If the degree of the numerator is greater than that of the denominator, the limit does not exist in a proper sense since the function increases or decreases without bound.
• If the degree of the numerator is less than that of the denominator, the limit at infinity is always \(0\), because the denominator outgrows the numerator.
• When the degrees are equal, the leading terms dominate, and the limit at infinity becomes the ratio of the leading coefficients, showing a horizontal asymptote at \(y = \frac{a_n}{b_n}\).

These rules help us predict the behavior of rational functions and identify any horizontal asymptotes, which further indicates the limiting behavior of the function graphically.

Polynomial Division

Polynomial division is a technique, similar to long division in numbers, that helps simplify expressions like rational functions. This process plays a pivotal role when evaluating limits of rational functions at infinity, especially when dealing with rational functions where the degrees of the numerator and the denominator are equal.

To divide one polynomial by another, we align the terms according to their degrees and proceed with division and subtraction steps repetitively. This approach helps in expressing a complex rational function in a more simplified manner which is often better for analytical purposes like finding limits or asymptotes.

Understanding polynomial division is vital because it allows us to isolate the key terms affecting the limit at infinity. By breaking down the polynomial expressions, we get an insightful view of the terms that vanish as \(x\) approaches infinity (often lesser degree terms), leaving us with the dominant leading term or constant for practical limit computations.

Leading Coefficients

Leading coefficients refer to the coefficients of the highest power of \(x\) present in a polynomial. In a polynomial \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0\), \(a_n\) is the leading coefficient.

Within the realm of rational functions, particularly when evaluating limits at infinity, the ratio of the leading coefficients becomes critical when the degrees of the numerator and denominator are equal. The limit as \(x\) approaches infinity for the rational function is simply the ratio \( \frac{a_n}{b_n} \) of these coefficients.

This concept simplifies the analytical work by focusing on these key coefficients as indicators of the horizontal asymptote. In other words, the bigger picture is dominated by these leading terms when \(x\) is sufficiently large, whereas lower-order terms diminish in impact.