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 can never cross one of its horizontal asymptotes. b. A rational function \(f\) has both \(\lim f(x)=L\) (where \(L\) is \(2+2\) finite) and \(\lim _{x \rightarrow-\infty} f(x)=\infty\) c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes. d. \(\lim _{x \rightarrow \infty}\left(x^{3}-x\right)=\lim _{x \rightarrow \infty} x^{3}-\lim _{x \rightarrow \infty} x=\infty-\infty=0\).

Short answer

b. Can a rational function have both a finite limit and an infinite limit as x approaches infinity? c. How many horizontal asymptotes can a function have at most? d. What is the limit as x approaches infinity of the given expression? a. No, the graph of a function can cross its horizontal asymptote an infinite number of times, but it is not guaranteed to always cross it. b. No, a rational function cannot have both a finite limit and an infinite limit as x approaches infinity. c. A function can have at most two horizontal asymptotes. d. The limit as x approaches infinity of the given expression is infinity.

Step-by-step solution

a. The graph of a function can never cross one of its horizontal asymptotes.

This statement is false. The graph of a function can cross its horizontal asymptote an infinite number of times. For example, consider the function \(f(x) = \frac{\sin x}{x}\). The horizontal asymptote for this function is \(y = 0\), and the graph of the function intersects this asymptote at every multiple of \(\pi\).

b. A rational function \(f\) has both \(\lim f(x)=L\) (where \(L\) is \(2+2\) finite) and \(\lim_{x \rightarrow -\infty} f(x)=\infty\)

This statement is false. For a rational function \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are non-zero polynomials, the value of the limit as \(x\) approaches infinity (or negative infinity) is determined by the highest degrees of \(P(x)\) and \(Q(x)\). Since the limit as \(x\) approaches negative infinity of \(f(x)\) is given as infinity, this means that the degree of \(P(x)\) must be greater than the degree of \(Q(x)\). Therefore, it is not possible for there to be a finite limit \(L\) as \(x\) approaches infinity for such a function.

c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.

This statement is true. A function can have any number of vertical asymptotes, depending on the roots of the denominator of a rational function. However, a function can have at most two horizontal asymptotes, one at the left end which is given by \(\lim_{x \rightarrow -\infty} f(x)\), and one at the right end given by \(\lim_{x \rightarrow \infty} f(x)\). Depending on the structure of the function, it may have zero, one, or two horizontal asymptotes.

d. \(\lim_{x \rightarrow \infty}(x^{3}-x)=\lim_{x \rightarrow \infty} x^{3}-\lim_{x \rightarrow \infty} x=\infty-\infty=0\).

This statement is false. First of all, the way in which the limits are subtracted is not valid for infinite limits. When dealing with infinite limit subtraction, we need to use the properties of limits to properly handle such expressions. In this particular case, we can factor out \(x\) from the expression to get \(\lim_{x \rightarrow \infty}(x^{3}-x) = \lim_{x \rightarrow \infty} x(x^{2}-1)\). As \(x \rightarrow \infty\), both \(x\) and \(x^2-1\) approach infinity, so the product would also approach infinity, not zero, as suggested. Thus, the correct answer is \(\lim_{x \rightarrow \infty}(x^{3}-x) = \infty\).

Key concepts

Limits of Rational Functions

Understanding the limits of rational functions is integral when analyzing their behavior as the input values approach certain points, including infinity. Simply put, a rational function is a fraction where both the numerator and denominator are polynomials. The limit of a rational function as the input variable, typically represented as `x`, approaches a particular value can inform us about the function's end behavior or the existence of any asymptotes.

Identifying Limit Behavior

Rational functions can exhibit different types of end behavior, where the terms with the highest powers of `x` in the numerator and denominator dominate. When the degree of the polynomial in the numerator is less than the degree in the denominator, the limit of the function as `x` approaches infinity is zero. Conversely, if the degree in the numerator is higher, the limit will approach infinity. For degrees that are equal, the limit as `x` approaches infinity is the ratio of the leading coefficients. Always remember, the end behavior reveals much about the horizontal asymptotes, which we'll discuss shortly.

When observing the step-by-step solution provided for the exercise in question, we improve understanding by considering the degrees of the polynomials in the numerator and denominator separately before jumping to conclusions about the limits. For both finite and infinite limits, students must apply limit properties correctly to avoid the misconception of the indeterminate form 'infinity minus infinity.'

Horizontal Asymptote Intersections

A horizontal asymptote represents a line that the graph of a function approaches as `x` moves towards positive or negative infinity. Students often mistakenly believe that a function's graph cannot cross a horizontal asymptote; however, this is not the case.

Intersections with Horizontal Asymptotes

Function graphs can, and often do, intersect their horizontal asymptotes. It's crucial to recognize that while an asymptote suggests a barrier the graph gets infinitely close to but does not cross, this 'rule' mainly applies to the tails of the graph as `x` goes towards infinity, not throughout its entire domain. For example, sinusoidal functions that are divided by `x` can cross their horizontal asymptote multiple times before 'settling down' as they head towards infinity. This behavior was highlighted in the solution to question a in the exercise provided, deterring the myth of asymptotic non-intersection.

Clarifying this concept fosters a deeper understanding that horizontal asymptotes are not an impenetrable boundary but rather a guideline for the function's long-term behavior. Additionally, the frequency and points of intersection with horizontal asymptotes can also give valuable insight into the function's periodic behavior or oscillations.

Vertical Asymptote Properties

Vertical asymptotes are lines that align with certain 'x' values where a function heads towards positive or negative infinity. They occur in rational functions where the denominator equals zero and the numerator is non-zero, indicating points of undefined values for the function.

Characteristics of Vertical Asymptotes

The positions of vertical asymptotes are determined by the roots of the denominator polynomial, as the function values become extremely large or small near these points. Notably, while a function can intersect a horizontal asymptote, it will never cross a vertical one. This is because a vertical asymptote corresponds to a 'discontinuity' in the function, which is a fundamental concept in calculus.

Understanding this property helps with sketching the graph of a rational function and predicting its behavior accurately. As confirmed in the solution to question c, rational functions can indeed have multiple vertical asymptotes, yet they are limited in the number of horizontal asymptotes they may possess. This distinction is rooted in the fundamentals of polynomial degrees, limits, and the definition of horizontal and vertical asymptotes respectively. Emphasizing these properties during instruction can assist students in avoiding the common pitfalls when dealing with complex rational functions.