What describes a vertical asymptote at x = a?

What describes a vertical asymptote at x = a?

• A. The function approaches a finite limit as x approaches a.
• B. The function equals f(a) at x = a.
• C. The function has a periodic behavior near a.
• D. The function becomes unbounded as x approaches a from one or both sides.

Answer: (D)

A vertical asymptote at x = a occurs when the function grows without bound as x gets arbitrarily close to a. In other words, the limit of f(x) as x approaches a is ±∞, possibly from one side or both sides. This means the values of the function blow up; it doesn’t settle to a finite number. For example, f(x) = 1/(x−a) has a vertical asymptote at x = a: as x approaches a from the right, f(x) → ∞, and as x approaches a from the left, f(x) → −∞. That unbounded behavior is the hallmark of a vertical asymptote. This differs from a finite limit, where f(x) would approach a specific finite value as x → a. It also differs from being defined at a (the function value at a might exist or not, but a vertical asymptote is about the behavior near a, not at the point itself). And it’s not about periodic behavior, which involves repeating patterns rather than unbounded growth near a. So the best description is that the function becomes unbounded as x approaches a from one or both sides.

A vertical asymptote at x = a occurs when the function grows without bound as x gets arbitrarily close to a. In other words, the limit of f(x) as x approaches a is ±∞, possibly from one side or both sides. This means the values of the function blow up; it doesn’t settle to a finite number.

For example, f(x) = 1/(x−a) has a vertical asymptote at x = a: as x approaches a from the right, f(x) → ∞, and as x approaches a from the left, f(x) → −∞. That unbounded behavior is the hallmark of a vertical asymptote.

This differs from a finite limit, where f(x) would approach a specific finite value as x → a. It also differs from being defined at a (the function value at a might exist or not, but a vertical asymptote is about the behavior near a, not at the point itself). And it’s not about periodic behavior, which involves repeating patterns rather than unbounded growth near a.

So the best description is that the function becomes unbounded as x approaches a from one or both sides.