Question

what is a vertical asymptote?

choose the correct answer below.

a. a vertical asymptote is a vertical line that the graph of the function approaches as x tends to positive or negative infinity.
b. a vertical asymptote is a vertical line \\(x = a\\), which the graph of a function approaches as \\(x\\) approaches \\(a\\).
c. a vertical asymptote is a vertical line \\(x = a\\), which the graph of a function departs from as \\(x\\) approaches \\(a\\).

Explanation:

Analyze the definition of a vertical asymptote

A vertical asymptote of a function \(y = f(x)\) is a vertical line \(x = a\) such that the function values \(f(x)\) approach positive or negative infinity as \(x\) approaches \(a\) from the left or from the right. Visually, the graph of the function approaches the vertical line \(x = a\) as \(x\) approaches \(a\).

Evaluate the given options

• Option A states that a vertical asymptote is approached as \(x\) tends to positive or negative infinity. This describes a horizontal asymptote, not a vertical asymptote.
• Option B states that a vertical asymptote is a vertical line \(x = a\), which the graph of a function approaches as \(x\) approaches \(a\). This correctly describes the behavior of a function near a vertical asymptote.
• Option C states that the graph "departs from" the line as \(x\) approaches \(a\), which is incorrect.

Answer:

• A. A vertical asymptote is a vertical line that the graph of the function approaches as x tends to positive or negative infinity.
• B. A vertical asymptote is a vertical line x = a, which the graph of a function approaches as x approaches a. (Correct answer)
• C. A vertical asymptote is a vertical line x = a, which the graph of a function departs from as x approaches a.