Problem 12 - 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} \left(\dfrac{1+5x}{1-4x} \right) \\]

Step-by-Step Solution

Answer

The limit as \(x\) approaches infinity of the function \( \frac{1+5x}{1-4x} \) is -1.25

Step 1: Identify the Degree of the Polynomial

Identify the degree of the numerator and the denominator. The degree of both the numerator and the denominator here is 1. The degree of the numerator is defined by the term with the highest power of \(x\), which is \(5x\). Similarly, the degree of the denominator is defined by the term with the highest power of \(x\), which is \(-4x\).

Step 2: Apply the Rule of Limits for Rational Functions

When the degrees of the numerator and the denominator are the same, the limit of the function as \(x\) approaches infinity is simply the ratio of the leading coefficients. The leading coefficient of the numerator is 5 and of the denominator is -4.

Step 3: Compute the Limit

As a result, \( \lim_{x\to \infty} \left(\frac{1+5x}{1-4x} \right) = \frac{5}{-4} = -1.25 \).

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients. In the simplest terms, they look something like this: \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). Here, \(a_n, a_{n-1}, \ldots, a_0\) are coefficients, and \(n\) is a non-negative integer representing the degree of the polynomial.

Polynomials can range from very simple, like \(x + 2\), to complex, like \(3x^3 + 2x^2 - x + 4\). They are foundational because they form the basis for more complex functions you'll encounter.

What's great about polynomial functions is they are continuous and smooth, meaning there are no breaks or jumps. Understanding these characteristics helps when moving to more complicated mathematical concepts.

Rational Functions

Rational functions are essentially a ratio (or fraction) of two polynomials. They have the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. Just like with other fractions, the denominator cannot be zero because division by zero is undefined.

As with our original exercise, handling rational functions often involves determining limits, particularly as variable \(x\) approaches infinity or a specific value. This helps in understanding the behavior of the function in different conditions.

Our example focuses on finding a limit as \(x\to\infty\). Such analysis can reveal the behavior of the function compared to the horizontal asymptotes, which inform where the function levels off on a graph.

Graphing Utility

A graphing utility is a tool that helps visualize mathematical functions. This can be a physical calculator or a software application like Desmos or GeoGebra.

These utilities are crucial because they allow you to see the graphical representation of equations, which can greatly aid in understanding. When verifying limits, graphing utilities can show you the trend of a function visually as \(x\to\infty\) or when approaching certain points.

Using a graphing utility helps confirm an analytical result. For instance, if the function levels out to a certain value as \(x\) becomes large, the graphing calculator will show this horizontal line, evidencing your computational work.

Degrees of Polynomials

The degree of a polynomial is determined by the highest power of the variable in the expression. It plays a crucial role in understanding the behavior of polynomial and rational functions.

For example, in the expression \(5x^2 + 3x + 1\), the degree is 2 because the highest power of \(x\) is 2. In rational functions, comparing the degree of the polynomial in the numerator and denominator helps in deducing limits.

If the degrees of both the numerator and the denominator are equal, as in the exercise displayed, you use the coefficients of their leading terms to find the limit. Understanding degrees is vital in predicting end behavior of functions, whether they rise, level off, or fall as \(x\to\infty\).