With a vertical asymptote at x = a, what happens to f(x) as x approaches a?

With a vertical asymptote at x = a, what happens to f(x) as x approaches a?

• A. f(x) approaches a finite limit.
• B. f(x) becomes unbounded toward ±∞.
• C. f(x) equals f(a).
• D. f(x) remains bounded away from a.

Answer: (B)

A vertical asymptote at x = a means the function’s values blow up without bound as x gets close to a. Near that point, f(x) grows in magnitude without settling to any finite number, so it heads toward infinity or negative infinity (often one side toward +∞ and the other toward −∞, though sometimes both sides head to the same sign). For instance, f(x) = 1/(x − a) becomes arbitrarily large in magnitude as x approaches a, with the sign depending on which side you approach from. This unbounded behavior is what characterizes the limit near a vertical asymptote.

A vertical asymptote at x = a means the function’s values blow up without bound as x gets close to a. Near that point, f(x) grows in magnitude without settling to any finite number, so it heads toward infinity or negative infinity (often one side toward +∞ and the other toward −∞, though sometimes both sides head to the same sign). For instance, f(x) = 1/(x − a) becomes arbitrarily large in magnitude as x approaches a, with the sign depending on which side you approach from. This unbounded behavior is what characterizes the limit near a vertical asymptote.