Question

use the graph to answer the question. which statement describes why ( x = -2 ) is an asymptote for the graphed function? (1 point) ( \bigcirc f(x)\to -2 ) as ( x\toinfty ) ( \bigcirc f(x)\toinfty ) as ( x\to -2 ) ( \bigcirc f(x)\to -infty ) as ( x\to -2 ) ( \bigcirc f(x)\to -2 ) as ( x\to -infty )

Explanation:

Step1: Recall the definition of a vertical asymptote

A vertical asymptote \(x = a\) occurs when \(f(x)\to\infty\) or \(f(x)\to-\infty\) as \(x\to a\).

Step2: Analyze each option

• For \(f(x)\to - 2\) as \(x\to\infty\) or \(x\to-\infty\), this describes a horizontal asymptote (not a vertical one).
• For \(x=-2\) to be a vertical asymptote, we need \(f(x)\to\pm\infty\) as \(x\to - 2\). Both \(f(x)\to\infty\) as \(x\to - 2\) and \(f(x)\to-\infty\) as \(x\to - 2\) imply \(x = - 2\) is a vertical asymptote.

Answer:

\(f(x)\to\infty\) as \(x\to - 2\) and \(f(x)\to-\infty\) as \(x\to - 2\) imply \(x=-2\) is a vertical asymptote. If we assume the function has a single - sided or two - sided infinite limit at \(x = - 2\), among the given options \(f(x)\to\infty\) as \(x\to - 2\) and \(f(x)\to-\infty\) as \(x\to - 2\) are the correct descriptions for \(x=-2\) being an asymptote. But if we consider the general form of vertical asymptote definition (either going to \(+\infty\) or \(-\infty\) as \(x\) approaches the value), both \(f(x)\to\infty\) as \(x\to - 2\) and \(f(x)\to-\infty\) as \(x\to - 2\) are valid. Since the options are separate, if the graph shows \(f(x)\to\infty\) as \(x\to - 2\) from one side and \(f(x)\to-\infty\) from the other (or one - sided infinite limit), the statements \(f(x)\to\infty\) as \(x\to - 2\) and \(f(x)\to-\infty\) as \(x\to - 2\) are correct for \(x = - 2\) being an asymptote.