asymptoten

An asymptote is a line that a curve approaches as the independent variable increases without bound or approaches a finite value. The concept captures the end behavior of curves and is used across algebra, calculus, and analytic geometry. Asymptotes are usually classified as vertical, horizontal, or oblique (slant).

Vertical asymptotes occur at finite x-values where the function becomes unbounded. Formally, x = a is a vertical asymptote if f(x) → ±∞ as x → a. A common example is f(x) = 1/(x − 2), which has a vertical asymptote at x = 2.

Horizontal asymptotes describe the limiting value of a function as x → ±∞. If f(x) → L, where L is a finite constant, then y = L is a horizontal asymptote. For instance, f(x) = (2x + 3)/(x + 1) tends to y = 2 as x → ±∞, giving a horizontal asymptote at y = 2. The simple function f(x) = 1/x has the horizontal asymptote y = 0.

Oblique (slant) asymptotes are straight lines of the form y = mx + b that the curve approaches as x → ±∞, with f(x) behaving like mx + b for large x. A criterion is lim_{x→∞} [f(x) − (mx + b)] = 0, where m = lim_{x→∞} f(x)/x and b = lim_{x→∞} [f(x) − mx]. An example is f(x) = x + 1/x, which has the oblique asymptote y = x.

Some curves have multiple asymptotes (for example, the hyperbola y = 1/x has x = 0 and y = 0 as asymptotes). Asymptotes describe long-run behavior rather than exact intersection points.