asymptote

Examples of asymptote in the following topics:

• Horizontal Asymptotes and Limits at Infinity

  • In the first case, $ƒ(x)$ has $y = c$ as asymptote when $x$ tends toward $- \infty$, and in the second that $ƒ(x)$ has $y = c$ as an asymptote as $x$ tends toward $+ \infty$.
  • More general type of asymptotes can be defined as the oblique asymptote case.
  • Only open curves that have some infinite branch, can have an asymptote.
  • No closed curve can have an asymptote.
  • The graph of a function can have two horizontal asymptotes.

• Asymptotes

  • A rational function can have at most one horizontal or oblique asymptote, and many possible vertical asymptotes; these can be calculated.
  • An asymptote that is neither horizontal or vertical is an oblique (or slant) asymptote.
  • A rational function has at most one horizontal or oblique asymptote, and possibly many vertical asymptotes.
  • If $n>m$, then there is no horizontal asymptote (However, if $n = m+1$, then there exists a slant asymptote).
  • Hence, horizontal asymptote is given by:

• Standard Equations of Hyperbolas

  • At large distances from the center, the hyperbola approaches two lines, its asymptotes, which intersect at the hyperbola's center.
  • A hyperbola approaches its asymptotes arbitrarily closely as the distance from its center increases, but it never intersects them.
  • Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±b⁄a, where b=a×tan(θ) and where θ is the angle between the transverse axis and either asymptote.
  • The distance b (not shown in below) is the length of the perpendicular segment from either vertex to the asymptotes.
  • If b = a, the angle 2θ between the asymptotes equals 90° and the hyperbola is said to be rectangular or equilateral.

• Parts of a Hyperbola

  • The asymptotes of the hyperbola are straight lines that are the diagonals of this rectangle.
  • Then draw in the asymptotes as extended lines that are also the diagonals of the rectangle.
  • Finally, draw the curve of the hyperbola by following the asymptote inwards, curving in to touch the vertex on the rectangle, and then following the other asymptote out.
  • The asymptotes of a rectangular hyperbola are the $x$- and $y$-axes.

• Infinite Limits

  • If the limit at infinity exists, it represents a horizontal asymptote at $y = L$.
  • Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.

• Curve Sketching

  • Determine the asymptotes of the curve.
  • Also determine from which side the curve approaches the asymptotes and where the asymptotes intersect the curve.

• Polynomial and Rational Functions as Models

  • They can take on only a limited number of shapes and are particularly ill-suited to modeling asymptotes.
  • For lots of datasets, their are no asymptotes and data is more or less bounded.
  • To deal with the asymptotic problems of polynomials, we also use rational functions:
  • Rational functions are a little more complex in form than polynomial functions, but they have an advantage in that they can take on a much greater range of shapes and can effectively model asymptotes.
  • However, rational functions sometimes include undesirable asymptotes that can disrupt an otherwise smooth trend line.

• Limited Growth

  • Graphically, the logistic function resembles an exponential function followed by a logarithmic function that approaches a horizontal asymptote.
  • This horizontal asymptote represents the carrying capacity.
  • That is, $y=c$ is a horizontal asymptote of the graph.
  • Additionally, $y=o$ is also a horizontal asymptote.
  • Logistic functions have an "s" shape, where the function starts from a certain point, increases, and then approaches an upper asymptote.

• Basics of Graphing Exponential Functions

  • That is, the curve approaches zero as $x$ approaches negative infinity making the $x$-axis is a horizontal asymptote of the function.
  • That is, the curve approaches zero as $x$ approaches negative infinity making the $x$-axis a horizontal asymptote of the function.
  • The function $y=b^x$ has the $x$-axis as a horizontal asymptote because the curve will always approach the $x$-axis as $x$ approaches either positive or negative infinity, but will never cross the axis as it will never be equal to zero.
  • The $x$-axis is a horizontal asymptote of the function.

• Graphs of Logarithmic Functions

  • The $y$-axis is a vertical asymptote of the graph.
  • This means that the $y$-axis is a vertical asymptote of the function.
  • However, the logarithmic function has a vertical asymptote descending towards $-\infty$ as $x$ approaches $0$, whereas the square root reaches a minimum $y$-value of $0$.
  • The graph of the square root function resembles the graph of the logarithmic function, but does not have a vertical asymptote.