Problem 29 - Precalculus with Limits

Question

In Exercises 29-34, use a graphing utility to graph the function and verify that the horizontal asymptote corresponds to the limit at infinity. $$ y = \dfrac{3x}{1-x} $$

Step-by-Step Solution

Answer

-3 is the horizontal asymptote.

Step 1: Identify and Graph the Function

Given function is \(y = \dfrac{3x}{1-x}\). To graph this function, a graphing tool or calculator can be used. The graph should include a curve that displays the function's behavior.

Step 2: Identify the horizontal asymptote

The horizontal asymptote of function \(y = \dfrac{3x}{1-x}\) is obtained when the function approaches infinity or negative infinity. In this case, as x approaches infinity, the term -x in the denominator will dominate, thus the function \(y = \dfrac{3x}{1-x}\) approaches -3 thus -3 is the horizontal asymptote.

Step 3: Verification

The graph will show a horizontal line y=-3 and the curve of the function will appear to approach this line as x increases or decreases without touching or crossing it, thus verifying that -3 is indeed the horizontal asymptote.

Key Concepts

Limits at Infinity

When we talk about limits at infinity, we refer to understanding the behavior of functions as the input, or x-value, grows larger and larger—specifically toward positive or negative infinity. This concept is crucial because it helps us determine the end behavior of a function, which in many cases leads us to discover horizontal asymptotes.
Take the function given in the exercise: \( y = \frac{3x}{1-x} \). Here, as \( x \) approaches positive infinity, \(-x\) in the denominator becomes very large, dominating the behavior of the function. To find the horizontal asymptote, we simplify the function:
\[ \lim_{{x \to \infty}} \frac{3x}{1-x} = \lim_{{x \to \infty}} \frac{3x}{-x} = -3 \]
This simplification reveals that the function's value approaches -3 as \( x \to \infty \). Therefore, \( y = -3 \) is the horizontal asymptote. This pattern repeats similarly if \( x \to -\infty \).

Understanding limits at infinity is essential in calculus as it explains how functions behave over extremely large values and helps us deduce horizontal asymptotes effectively.

Graphing Utility

Graphing utilities are tools that assist us in visualizing functions by plotting their graphs, which can be invaluable when trying to understand complex behavior or features like asymptotes. Whether it's a graphing calculator or software, these utilities allow students and mathematicians to input functions and instantly view their graphs.

In the case of the function \( y = \frac{3x}{1-x} \), a graphing utility can clearly depict the curve of the function and how it interacts with its horizontal asymptote \( y = -3 \). By simply entering this function into a graphing tool, we can observe:

• The curve approaching the line \( y = -3 \).
• The behavior of the function as \( x \) increases or decreases significantly, showing the limits at infinity becoming apparent.
• Any points of intersection, discontinuities, or other traits.

Using a graphing utility not only confirms theoretical calculations but can also demonstrate the practical visual aspect of mathematical concepts, making them easier to grasp and analyze.

Rational Functions

Rational functions are a significant class of functions in mathematics, defined as the ratio of two polynomials. The function \( y = \frac{3x}{1-x} \) is a simple example of a rational function.

Rational functions can have distinct characteristics such as:

• Horizontal and vertical asymptotes - found by analyzing the limits of the function and when the denominator equals zero, respectively.
• Intercepts - points where the function crosses the x-axis or y-axis.
• End behavior - determined by the degrees of the polynomials in the numerator and denominator.

For \( y = \frac{3x}{1-x} \), the horizontal asymptote was determined to be \( y = -3 \) by analyzing the limits at infinity. Rational functions often have their end behavior dictated by the leading coefficients and degrees of the polynomials. Here, since both polynomials are of the same degree, the limit at infinity results in a horizontal asymptote defined by the ratio of their coefficients.

Understanding rational functions is key for students because they encompass a wide variety of problems across calculus and algebra, showcasing behaviors that depart from simple lines and parabolas.