Question

If, as \(x \rightarrow-\infty\) or as \(x \rightarrow \infty,\) the values of \(R(x)\) approach some fixed number \(L,\) then the line \(y=L\) is a ________ ___________ of the graph of R.

Short answer

horizontal asymptote.

Step-by-step solution

- Understand the Terminology

First, identify what it means for the values of a function, in this case, \(R(x)\), to approach a fixed number \(L\) as \(x\) approaches positive or negative infinity. This concept refers to the function approaching a horizontal line at infinity.

- Identify the Definition

Next, recognize from the definition of asymptotes that if a function approaches a fixed number \(L\) as \(x\) goes to \(+\text{infinity}\) or \(-\text{infinity}\), then the line \(y = L\) is considered a horizontal asymptote.

- Apply the Definition

Apply the understanding from the previous steps to the given function \(R(x)\). Since \(R(x)\) approaches \(L\) as \(x\) goes to \(+\text{infinity}\) or \(-\text{infinity}\), the line \(y = L\) must be a horizontal asymptote.

Key concepts

Limits at Infinity

When we talk about limits at infinity, we're discussing the behavior of a function as the input, usually denoted as \(x\), grows very large in the positive or negative direction. If \(R(x)\) approaches a fixed number \(L\) as \(x\) goes to positive or negative infinity, we write this symbolically as:
\[ \lim_{{x \to \rightarrow \text{+\textinfinity} }} R(x) = L \] or
\[ \lim_{{x \to \rightarrow -\text{+\textinfinity} }} R(x) = L \] This notation simply means that as \(x\) becomes extremely large in either direction, the values of \(R(x)\) get closer and closer to the number \(L\). This concept is foundational in understanding horizontal asymptotes because it describes how a function behaves far away from the origin. Understanding these limits helps in visualizing the end behavior of functions.

Behavior of Rational Functions

Rational functions are fractions where both the numerator and denominator are polynomials. These functions are essential to study because their end behavior often involves horizontal asymptotes. Consider a rational function \(R(x) = \frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. The degree of the polynomials (the highest power of \(x\)) plays a crucial role in determining the behavior of the function. Here's what happens as \(x\) approaches infinity or negative infinity:

• If the degree of \(P(x)\) is less than the degree of \(Q(x)\), \(R(x)\) approaches 0.
• If the degree of \(P(x)\) is equal to the degree of \(Q(x)\), \(R(x)\) approaches the ratio of the leading coefficients of \(P(x)\) and \(Q(x)\).
• If the degree of \(P(x)\) is greater than the degree of \(Q(x)\), \(R(x)\) does not approach a finite limit (horizontal asymptote).

Understanding this behavior helps in visualizing how the function behaves at extreme values of \(x\) and identifying any horizontal asymptotes.

Asymptotic Behavior

The term 'asymptotic behavior' refers to the tendency of a function to get closer and closer to a particular line (or curve) as the input grows very large or very small. In the context of our discussion, we focus on horizontal asymptotes. A horizontal asymptote is a horizontal line that the graph of the function approaches but never actually reaches as \(x\) goes to positive or negative infinity. Symbolically, if \(y = L\) is a horizontal asymptote for the function \(R(x)\), then either
\[ \lim_{{x \to \rightarrow \text{+\textinfinity} }} R(x) = L \] or
\[ \lim_{{x \to \rightarrow -\text{+\textinfinity} }} R(x) = L \] This means that no matter how large or small \(x\) gets, the function \(R(x)\) will get as close as we want to the line \(y = L\). However, it never actually touches or crosses this line at infinity. This concept helps us understand how functions behave over very large intervals and is crucial in many areas of calculus and advanced algebra.