Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The graph of a function never crosses one of its horizontal asymptotes. b. A rational function \(f\) has both \(\lim f(x)=L\) (where \(L\) is finite) and \(\lim _{x \rightarrow-\infty} f(x)=\infty\). \(^{x \rightarrow 2}\) c. The graph of a function can have at most two horizontal asymptotes.

Short answer

Question: Determine whether each statement is true or false, and provide an explanation or counterexample where necessary. a) The graph of a function never crosses one of its horizontal asymptotes. b) A rational function can have a finite limit L and an infinite limit as x approaches -∞. c) The graph of a function can have at most two horizontal asymptotes.

Step-by-step solution

Statement (a) Evaluation

We need to determine if the graph of a function never crosses one of its horizontal asymptotes. The statement is false. There are functions whose graph can cross its horizontal asymptote. For example, let's consider the function \(f(x) = \frac{\sin x}{x}\). The limit as \(x \rightarrow \infty\) is \(0\), so \(y = 0\) is a horizontal asymptote, but the graph of the function intersects the line \(y = 0\) infinitely many times because \(\sin x = 0\) whenever \(x\) is an integer multiple of \(\pi\).

Statement (b) Evaluation

We need to determine if a rational function \(f\) can have a finite limit \(L\) and an infinite limit as \(x\) approaches \(-\infty\). The statement is true. Let's consider a rational function \(f(x) = \frac{x^2 + 1}{x^2 - 4}\). Now, let's find its limits. - As \(x \rightarrow 2\), we have: \(\lim_{x \rightarrow 2} f(x) = \lim_{x \rightarrow 2} \frac{x^2 + 1}{x^2 - 4} = \frac{5}{0}\) which is not finite. - As \(x \rightarrow -\infty\), we have: \(\lim_{x \rightarrow -\infty} f(x) = \lim_{x \rightarrow -\infty} \frac{x^2 + 1}{x^2 - 4} = 1\) which is finite. So, statement (b) is true because we found a rational function that has both finite and infinite limits.

Statement (c) Evaluation

We need to determine if the graph of a function can have at most two horizontal asymptotes. The statement is true. A function can have at most two horizontal asymptotes. These asymptotes occur in the limits as \(x \rightarrow \infty\) and \(x \rightarrow -\infty\). A function cannot have more than two horizontal asymptotes because it would imply that the function's behavior is changing drastically at different infinities, which is not possible for a single-variable function.

Key concepts

Limit of a Function

Understanding the concept of a limit of a function is crucial for analyzing the behavior of functions, especially as they approach infinity or a specific point. The limit of a function describes the value that the function approaches as the input (or 'x' value) gets closer to a given number.

For example, let's examine the scenario where we approach a specific point, such as x = 2. If a function f(x) gets closer and closer to a number L as x approaches 2, we say that the limit of f(x) as x approaches 2 is L, written as \( \text{lim}_{x \rightarrow 2} f(x) = L \). It's important to note that while a function may never actually reach this value, it's about the behavior as it gets infinitely close.

Contrarily, if we consider the function's behavior at the borders of its domain, such as \( \text{lim}_{x \rightarrow \infty} f(x) \) or \( \text{lim}_{x \rightarrow -\infty} f(x) \), we are discussing horizontal asymptotes. These are lines that the graph of the function approaches but does not necessarily touch or cross as x goes to infinity (positive or negative).

It is a common misconception that a function cannot cross a horizontal asymptote. In reality, the definition of a horizontal asymptote does not prevent the function from intersecting the asymptote at some points; it is wholly concerned with the end behavior of the function as x approaches infinity.

Rational Functions

A rational function is defined as the ratio of two polynomials, expressed as \( f(x) = \frac{P(x)}{Q(x)} \) where both P(x) and Q(x) are polynomials and Q(x) is not equal to zero. The limits of rational functions as x approaches infinity or a specific value can be finite or infinite, depending on the behavior of the numerator and denominator.

For instance, consider the function \( f(x) = \frac{x^2 + 1}{x^2 - 4} \). If we look at the limit as x approaches 2, the denominator approaches zero, causing the value of the function to increase without bound, theoretically approaching infinity. Thus, in this case, the limit is not finite. But when x approaches negative infinity, the highest powers of x in both the numerator and the denominator dictate the limit, and because they are the same (x^2), the function approaches a finite limit, in this case, 1.

Understanding these concepts helps explain why statement (b) from the exercise is true, demonstrating that rational functions can indeed have different behaviors at different values or at infinity. They can approach a finite value or become unbounded.

Graph Behavior

The graph behavior of functions, particularly near their boundaries or asymptotes, is a key aspect of calculus and higher-level mathematics. The horizontal asymptote is a concept tied closely to the end behavior of functions. As previously discussed, it is a common misconception that a function cannot have more than one horizontal asymptote or that it can't cross its horizontal asymptote. However, the behavior of functions differs greatly.

A function can indeed have two horizontal asymptotes—one as x approaches positive infinity and one as x approaches negative infinity. This usually happens for rational functions where the degrees of the numerator and denominator polynomials are the same. The coefficients of the highest-degree terms in both the numerator and the denominator dictate the horizontal asymptotes.

Moreover, the possible intersections with these asymptotes tell us that while the function may get infinitely close to the asymptote line, it can still cross or touch it at various points. This understanding debunks statement (a) from the exercise and aligns with statement (c), which states that a function can have at most two horizontal asymptotes. These principles are key to predicting the long-term behavior of a function and determining its overall graph shape.