Question

Determine whether the following statements are true and give an explanation or counterexample. a. The graph of a function can never cross one of its horizontal asymptotes. b. A rational function \(f\) can have both \(\lim _{x \rightarrow \infty} f(x)=L\) and \(\lim _{x \rightarrow-\infty} f(x)=\infty\). c. The graph of any function can have at most two horizontal asymptotes.

Short answer

Based on the step-by-step solution: (a) A graph can cross a horizontal asymptote only at a finite number of points. [False] Explanation: Consider the function \(f(x) = x \sin(1/x)\) for \(x \neq 0\), and \(f(x) = 0\) for x = 0. The graph of this function will cross the horizontal asymptote at y = 0 infinitely many times as x approaches 0. (b) There exists a rational function \(f\) such that \(\lim _{x \rightarrow \infty} f(x)=L\) and \(\lim _{x \rightarrow -\infty} f(x)=\infty\). [False] Explanation: While there can be a rational function where both limits approach a finite number or infinity, there is no rational function where one limit approaches a finite number (L) and the other approaches infinity. (c) A graph can have one or two horizontal asymptotes, depending on the limiting values as \(x \rightarrow \infty\) and \(x \rightarrow -\infty\). [True] Explanation: A graph can have at most two horizontal asymptotes, one as \(x \rightarrow \infty\) and one as \(x \rightarrow -\infty\). If a graph approaches the same limit as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\), then it has only one horizontal asymptote. If it approaches different limits, it has two horizontal asymptotes.

Step-by-step solution

Statement (a) Explanation or Counterexample

Consider the function \(f(x) = x \sin(1/x)\) for \(x \neq 0\), and \(f(x) = 0\) for x = 0. The graph of this function will cross the horizontal asymptote at y = 0 infinitely many times as x approaches 0. Therefore, the statement is false.

Statement (b) Explanation or Counterexample

Consider the rational function \(f(x)=\frac{1}{x}\). We have \(\lim _{x \rightarrow \infty} f(x) = 0\) and \(\lim _{x \rightarrow -\infty} f(x) = 0\), showing that a rational function \(f\) can indeed have both \(\lim _{x \rightarrow \infty} f(x)=L\) and \(\lim _{x \rightarrow -\infty} f(x)=L\). However, it is not possible to find a rational function that has \(\lim _{x \rightarrow \infty} f(x)=L\) and \(\lim _{x \rightarrow -\infty} f(x)=\infty\). To understand this, we can reason that rational functions are in the form of \(f(x) = \frac{p(x)}{q(x)}\), where p(x) and q(x) are polynomials. As x approaches infinity or negative infinity, we only have the possibilities of having L to be a finite number, infinity or negative infinity. Therefore, the statement is false.

Statement (c) Explanation or Counterexample

A graph can have at most two horizontal asymptotes, one as \(x \rightarrow \infty\) and one as \(x \rightarrow -\infty\). If a graph approaches a limit L as \(x \rightarrow \infty\) and a different limit M as \(x \rightarrow -\infty\), then it has two horizontal asymptotes. If a graph approaches the same limit L as \(x \rightarrow \infty\) and as \(x \rightarrow -\infty\), then it has only one horizontal asymptote. The statement is true.

Key concepts

Functions

In mathematics, a function describes a relationship between a set of inputs and a set of possible outputs. Each input in a function corresponds to exactly one output. This concept is foundational in calculus and helps in understanding more complex ideas such as limits and asymptotes. Functions are generally represented by symbols like \(f(x)\), which means "the function of \(x\)."

• **Domain:** All the inputs a function can accept.
• **Range:** All possible outputs a function can produce.

Functions can be as simple as \(f(x) = x + 2\), linear functions, or more complicated, such as rational functions or trigonometric functions. Understanding functions involves analyzing their behavior, such as where they increase or decrease, and whether they have any asymptotes, zeros, or maximum and minimum points.

Limits

The concept of limits is about finding the value that a function approaches as the input approaches a particular point. This is a crucial building block in calculus because it leads to the formal definitions of derivatives and integrals. For example, as \(x\) approaches a certain number, the function \(f(x)\) might approach a specific value, which is its limit. This is written mathematically as \(\lim_{x \to a} f(x) = L\).

Here, \(a\) represents the input value \(x\) is approaching, and \(L\) is the limit value. Limits help us understand the concept of continuity and can be used to find asymptotes of functions, as shown in our exercise above. They also solve problems where functions involve division by zero or very large numbers.

Asymptotes

Asymptotes are lines that a graph of a function approaches but never quite touches. They help describe the behavior of functions as the inputs become very large or very small. We typically encounter horizontal, vertical, and occasionally oblique asymptotes in calculus.

• **Horizontal asymptotes** occur when \(y\) approaches a constant as \(x\) goes to infinity or negative infinity.
• **Vertical asymptotes** occur when \(x\) approaches a certain value that makes the function undefined, thus causing \(y\) to approach infinity.

In the exercise example, the statement "the graph can never cross its horizontal asymptote" is false because some functions can indeed cross their horizontal asymptotes. Understanding asymptotes involves analyzing limits and can give insight into a function's long-term behavior.

Rational Functions

Rational functions are fractions made up of polynomials and are written in the form \(f(x) = \frac{p(x)}{q(x)}\). These functions are particularly interesting because they can have asymptotes. The behavior of rational functions as \(x\) approaches infinity or negative infinity is highly dependent on the degrees of the polynomials in both the numerator and denominator.

• If the degree of \(p(x)\) is less than \(q(x)\), the function has a horizontal asymptote at \(y = 0\).
• If the degrees are equal, the horizontal asymptote is \(y = \frac{leading \ coefficient \ of \ p(x)}{leading \ coefficient \ of \ q(x)}\).
• If the degree of \(p(x)\) is greater than \(q(x)\), there is no horizontal asymptote.

Rational functions cannot have one finite limit as \(x\) approaches positive infinity and a differing infinite limit as \(x\) approaches negative infinity, which contradicts one of the statements in the exercise. They provide a rich field of study in calculus due to their complex behavior and connection to real-world phenomena.