Asymptotes in a sentence

A hyperbola approaches its asymptotes arbitrarily closely as the distance from its center increases, but it never intersects them; however, a degenerate hyperbola consists only of its asymptotes.

Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph.

Asymptotes for rational functions A rational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.

Horizontal asymptotes The graph of a function can have two horizontal asymptotes.

A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.

Conic section analysis of the hyperbolic appearance of circles Figure 10: The hyperbola as a circle on the ground seen in perspective while gazing down slightly, showing the circle's (non-parallel) tangents as asymptotes.

Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions.

Further elevation of the gaze into the sky then brings the point of convergence of the asymptotes towards the viewer.

Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or ∞.

If one forms a rectangle with vertices on the asymptotes and two sides that are tangent to the hyperbola, the sides tangent to the hyperbola are 2b in length while the sides that run parallel to the line between the foci (the major axis) are 2a in length.

If the curve has a singular point at infinity which may have several asymptotes or parabolic branches.

More general type of asymptotes can be defined in this case.

Oblique asymptotes are diagonal lines so that the difference between the curve and the line approaches 0 as x tends to +∞ or ∞.

On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (the asymptotes of the function), they intersect at exactly the cutoff frequency.

Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have several asymptotes or parabolic branches.

So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch.

The distance b (not shown) is the length of the perpendicular segment from either vertex to the asymptotes.

These hyperbolas are described by the equation : where the foci are located at a distance c from the origin on the x-axis, and where θ is the angle of the asymptotes with the x-axis.

The solution to this is : In this case, the curve has vertical asymptotes and this limits the span to πc.

Thus, both the x and y-axes are asymptotes of the curve.