Question

which term below correctly completes the following sentence?
near a functions vertical asymptotes, its values become very ______ positive
numbers or negative numbers with very large absolute values.
a. large
b. small
c. undefined
d. rational

Explanation:

A vertical asymptote of a function \(y = f(x)\) is a vertical line \(x = a\) such that as \(x\) approaches \(a\) (from the left or the right), the function values \(y=f(x)\) approach \(\pm\infty\). When \(y = f(x)\to\infty\), the function values are very large positive numbers. When \(y = f(x)\to-\infty\), the function values are negative numbers with very large absolute values.

• Option B: If the values were very small, as \(x\) approaches the vertical asymptote, \(y\) would approach \(0\), which is not the case for vertical asymptotes.
• Option C: The function is undefined at the vertical asymptote (the value \(x = a\) itself), but the question is about the values of the function near (not at) the vertical asymptote.
• Option D: A rational number is a number that can be expressed as \(\frac{p}{q}\) where \(p,q\in\mathbb{Z}\) and \(q

eq0\). This has no relation to the behavior of a function near a vertical asymptote.

Answer:

A. large