Problem 14 - Precalculus with Limits

Question

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \dfrac{1-2x}{x+2} \\]

Step-by-Step Solution

Answer

The limit of the given function as x approaches infinity is -2.

Step 1: Identify the Degree of the Polynomial in both Numerator and Denominator

Looking at the given function, the polynomial in the numerator is \(1-2x\) of degree 1 and the one in the denominator is \(x+2\) also of degree 1. They both have the same degree.

Step 2: Apply Rule for Evaluating Limits at Infinity for Rational Functions with Same Degree

The limit of a rational function as x approaches infinity, where the numerator and the denominator have the same degree, can be simply evaluated by taking the ratio of the leading coefficients. In this case, we have a leading coefficient of -2 in the numerator and 1 in the denominator.

Step 3: Evaluate the Limit

The limit of the given function as x approaches infinity, given by the ratio of the leading coefficients, is \(-2/1=-2\).

Key Concepts

Polynomial Degree

The degree of a polynomial is a vital concept in calculus and algebra. It tells us the highest power of the variable in the polynomial expression. This is significant when evaluating limits of rational functions. For instance, in the polynomial \(1 - 2x\), the highest power of \(x\) is 1. Therefore, its degree is 1. Similarly, in \(x + 2\), the highest power is also 1. Understanding polynomial degree helps us determine how polynomial expressions behave as the variable grows very large, particularly when finding limits at infinity.

Rational Function

Rational functions are fractions where the numerator and the denominator are polynomials. They play a critical role in calculus, especially when discussing limits. When you have a rational function like \(\frac{1-2x}{x+2}\), you can quickly investigate how it behaves for large values by analyzing its structure. To find the limit at infinity, recognizing that a rational function's behavior is often dominated by its highest degree terms is key. Rational functions often approach a finite limit, rise or fall to infinity, or oscillate, depending on the degrees of their numerator and denominator polynomials.

Limits at Infinity

Limits at infinity focus on understanding the behavior of functions as the variable grows larger and larger without bound. They are crucial in analyzing the end behavior of rational functions. For rational functions, limits at infinity can be computed by focusing on their leading terms because they largely determine the function's growth rates. By studying these limits, we can make conclusions about trends: whether the function approaches a flat line, grows indefinitely, or something else entirely. In the given example, we evaluate the limit \(\lim_{x\to \infty} \frac{1-2x}{x+2}\) by identifying degrees and using the leading coefficients for a sound conclusion.

Leading Coefficients

Leading coefficients are the coefficients of the term with the highest degree in a polynomial. They are instrumental in determining limits at infinity of rational functions with identical degrees in the numerator and denominator. When these degrees match, the limit is determined simply by the ratio of these leading coefficients. For example, in \(1-2x\), the leading coefficient is \(-2\). Similarly, in \(x+2\), it is \(1\). Thus, the limit at infinity for the function \(\frac{1-2x}{x+2}\) is calculated as \(\frac{-2}{1} = -2\). This demonstrates the potency of leading coefficients in simplifying complex limit evaluations.