asymptotes

An asymptote is a line that a curve approaches as the independent variable grows without bound or as it approaches a finite singularity. In the plane, the common types are vertical, horizontal, and oblique (slant) asymptotes.

Vertical asymptote: A vertical line x = a is a vertical asymptote of a function f if either lim_{x→a^+} f(x) = ±∞ or lim_{x→a^-} f(x) = ±∞. The graph nears the line x = a from either side but does not reach it in the limit.

Horizontal asymptote: A horizontal line y = b is a horizontal asymptote if lim_{x→±∞} f(x) = b. This describes end behavior as x grows large in magnitude.

Oblique (slant) asymptote: A line y = mx + b is an oblique asymptote if lim_{x→±∞} [f(x) − (mx + b)] = 0. This occurs when the growth of f(x) is essentially linear for large |x|; for example, f(x) = 2x + 3 + 1/x has the oblique asymptote y = 2x + 3.

Rational functions provide classic cases: if deg(numerator) = deg(denominator), the horizontal asymptote is y = leading coefficient ratio; if deg(numerator) = deg(denominator) + 1, there is a slant asymptote given by the leading terms.

Examples: f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. f(x) = (2x^2 + 3x + 1)/x has oblique asymptote y = 2x + 3.

Some curves have no asymptotes (e.g., a circle). Others may have multiple branches, each with its own asymptote.