Examples for

Asymptotes

An asymptote is a line that a curve becomes arbitrarily close to as a coordinate tends to infinity. The simplest asymptotes are horizontal and vertical. In these cases, a curve can be closely approximated by a horizontal or vertical line somewhere in the plane. Some curves, such as rational functions and hyperbolas, can have slant, or oblique, asymptotes, which means that some sections of the curve are well approximated by a slanted line.

Asymptotes

Find horizontal, vertical and slant asymptotes of rational functions and more.

Compute asymptotes of a function:

asymptotes (2x^3 + 4x^2 - 9)/(3 - x^2)
asymptotes of erf(x)

Find asymptotes of a curve given by an equation:

asymptotes x^2 + y^3 = (x y)^2

Horizontal Asymptotes

Find horizontal asymptotes of a function.

Compute horizontal asymptotes:

horizontal asymptotes
horizontal asymptotes -2x / sqrt(1+x^2)
horizontal asymptotes tanh(x^2)

Vertical Asymptotes

Compute vertical asymptotes of an expression.

Compute vertical asymptotes:

vertical asymptotes
vertical asymptotes cot(x)
vertical asymptotes (x^5 - 12x^3 + 9x)/(x^3 - 4x)

Oblique Asymptotes

Calculate slant, or oblique, asymptotes.

Compute oblique asymptotes:

slant asymptote
oblique asymptotes (4x^3 + 1)/(x^2 - 1)

RELATED EXAMPLES

  • Algebra
  • Limits
  • Polynomials
  • Rational Functions

    • Curvilinear Asymptotes

    Find parabolic and other curvilinear asymptotes.

    Compute polynomial asymptotes of a rational function:

    curvilinear asymptote
    polynomial asymptotes (x^12 - x^6 + 1)/(x^8 - 16x^4 + 4)