Question

True or False If \(x=3\) is a vertical asymptote of the graph of a rational function \(R,\) then as \(x \rightarrow 3,|R(x)| \rightarrow \infty\).

Short answer

True

Step-by-step solution

Understand the definition of a vertical asymptote

A vertical asymptote of a function occurs when the function's value approaches \(\text{either positive or negative infinity}\) as the input approaches a specific value from either the left or the right. For a rational function \( R(x) = \frac{P(x)}{Q(x)} \), where \(P(x) \) and \(Q(x)\) are polynomials, a vertical asymptote occurs when \( Q(x)=0 \) and \(P(x) \) does not equal zero at \(x = a\).

Identify the given conditions

It is given that \(x = 3\) is a vertical asymptote of the rational function \(R(x)\). This means that as \(x \rightarrow 3\), the function \(R(x)\) will tend towards \( \text{either positive or negative infinity} \), based on whether the function values on either side of \(x = 3\) approach very large positive or negative values.

Determine the implication of the asymptote

Since \(x = 3\) is a vertical asymptote, by definition, as \(x \rightarrow 3 \) from the left (\( x \rightarrow 3^-\)) or from the right (\( x \rightarrow 3^+ \)), the magnitude \( |R(x)| \) must approach infinity. This implicates that \( |R(x)| \rightarrow \ \text{infinity} \) as \ x \ approaches \ 3 \.

Conclude True or False

Given that \(x=3\) is a vertical asymptote, indeed by definition, \( |R(x)| \rightarrow \ \text{infinity} \) as \ x \ approaches \ 3 \. Therefore, the statement is True.

Key concepts

rational function

A rational function is a type of function that can be expressed as the ratio of two polynomials. In other words, if you have two polynomial expressions, say, \(P(x)\) and \(Q(x)\), a rational function \(R(x)\) is defined as:

• \(R(x) = \frac{P(x)}{Q(x)}\)

The numerator \(P(x)\) and the denominator \(Q(x)\) can be any polynomial of any degree.

For example, \(R(x) = \frac{2x^2 + 3x - 5}{x^2 - 1}\) is a rational function where:

• \(P(x) = 2x^2 + 3x - 5\)
• \(Q(x) = x^2 - 1\)

Rational functions are very important in mathematics because they can model a wide variety of real-world phenomena. However, when the denominator \(Q(x)\) is zero, the function is undefined, often leading to vertical asymptotes.

asymptote definition

An asymptote is a line that a graph approaches but never actually touches. There are three main kinds of asymptotes: vertical, horizontal, and oblique. Here, we are focusing on vertical asymptotes.

A vertical asymptote occurs when the function values grow without bound as the input approaches a certain value. For a function \(R(x)\), this usually happens when the denominator \(Q(x)\) becomes zero while the numerator \(P(x)\) is non-zero:

• For example, if \(Q(x) = x - 3\), so at \(x = 3\), the function has a vertical asymptote.

As you approach the vertical asymptote, the function values can either go to positive infinity, negative infinity, or both.

Understanding this helps us see why the given statement is true. A vertical asymptote at \(x = 3\) means as \(x\) gets closer to 3 from either side, \(R(x)\) increases or decreases without limit.

limits

Limits are fundamental in calculus and help us understand the behavior of functions at specific points, especially where they may not be defined. The limit of a function \(R(x)\) as \(x\) approaches a specific value \(a\) is written as:

\[ \lim_{{x \to a}} R(x) \]

When dealing with vertical asymptotes, limits tell us how the function behaves as it gets closer to that asymptote. For a vertical asymptote at \(x = 3\), we look at:

• \(\lim_{{x \to 3^-}} R(x)\)
• \(\lim_{{x \to 3^+}} R(x)\)

This will show if the function approaches positive or negative infinity as \(x\) approaches 3 from the left and the right.

In the given exercise, since \( x = 3 \) is a vertical asymptote of \(R(x)\), we conclude that:

• \(|R(x)| \to \infty\) as \(x \to 3\)

This confirms the statement is true. Understanding limits is key to mastering concepts involving vertical asymptotes and rational functions.