Problem 2 - Algebra and Trigonometry

Question

If the rational function \(y=r(x)\) has the horizontal asymptote \(y=2,\) then \(y \rightarrow\) ________ as \(X \rightarrow \pm \infty\)

Step-by-Step Solution

Answer

2

Step 1: Understanding Asymptotes

A horizontal asymptote of a function describes the behavior of a function as the input variable grows very large or very small. For rational functions, a horizontal asymptote gives us the value that the function approaches as \(x\) approaches positive or negative infinity.

Step 2: Identifying Horizontal Asymptote

We are given that the rational function \(y = r(x)\) has a horizontal asymptote at \(y = 2\). This means that as \(x\) approaches either positive infinity or negative infinity, the value of \(y\) will approach 2.

Step 3: Completing the Expression

Thus, based on the behavior of rational functions and the given horizontal asymptote, we conclude: \(y \rightarrow 2\) as \(x \rightarrow \pm \infty\).

Key Concepts

Rational Functions

Rational functions are expressions of the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). These functions are like fractions, where the numerator and denominator are both polynomials.

Understanding rational functions is crucial as they often exhibit interesting behavior at large values of \( x \). These behaviors can include horizontal asymptotes, which we will discuss more later. Because rational functions consist of polynomials, their growth rates largely depend on the degrees of these polynomials.

Here's a quick breakdown of growth approaching infinity:

• If the degree (highest power of \( x \)) of \( P(x) \) is greater than that of \( Q(x) \), the function increases or decreases without bound as \( x \) approaches infinity.
• If the degrees of \( P(x) \) and \( Q(x) \) are the same, the horizontal asymptote is determined by the leading coefficients of them.
• If the degree of \( P(x) \) is less than that of \( Q(x) \), the function typically approaches zero as \( x \) goes to infinity.

These rules help in understanding the long-term behavior of rational functions.

Limits at Infinity

Limits at infinity are a way to describe what happens to a function's output as the input variable becomes very large—either positively or negatively. Simply put, they tell us the end behavior of a function. For rational functions, the limit as \( x \rightarrow \infty \) or \( x \rightarrow -\infty \) is often tied to the concept of horizontal asymptotes.

When dealing with limits at infinity, you might find an expression like \( \lim_{x \to \infty} f(x) = L \), which means as \( x \) grows larger and larger, the function \( f(x) \) approaches the value \( L \). This same principle applies as \( x \to -\infty \).

Let's connect this to rational functions:

• If \( y = r(x) \) has a horizontal asymptote, the limit at infinity of \( r(x) \) would be the value of the asymptote.
• For example, if \( y = r(x) \) has a horizontal asymptote at \( y = 2 \), then \( \lim_{x \to \pm \infty} r(x) = 2 \).

These limits give a very clear picture of how the function behaves at extreme ranges of the variable.

Asymptotic Behavior

The asymptotic behavior of a function describes how it behaves as it approaches infinity or a certain point. For rational functions, especially in this context, we're mostly concerned with horizontal asymptotes. A horizontal asymptote is the value that a function approaches, but never quite reaches, as \( x \) tends toward infinity or negative infinity.

To determine if a rational function has a horizontal asymptote, assess the degrees of the polynomial in the numerator and the denominator:

• As mentioned, if their degrees are equivalent, the horizontal asymptote can be found by dividing the leading coefficients.
• If the numerator's degree is lower, the horizontal asymptote is usually \( y = 0 \).

Returning to our original exercise, the function with a horizontal asymptote of \( y = 2 \), tells us that both as \( x \) approaches \( +\infty \) and \( -\infty \), the output of the function gradually approaches 2.

This offers a powerful insight into the "end" behavior of a function, helping us graph and understand how it interacts with its surrounding space.