the asymptotes of a hyperbola
Sketch a hyperbola and you will notice its two branches straighten out as they fly away from the center, each branch hugging a fixed straight line ever more tightly without ever reaching it. Those two guide lines are the asymptotes, and they are the single most useful aid for drawing a hyperbola accurately.
For the standard hyperbola x^2/a^2 - y^2/b^2 = 1, the asymptotes are the two lines y = (b/a)x and y = -(b/a)x, crossing at the center. Here is why: solve the equation for y to get y = +-(b/a) sqrt(x^2 - a^2); as x grows huge, the a^2 under the root becomes negligible, so y approaches +-(b/a)x. The gap between curve and line shrinks to zero but never closes. A clean way to draw them: mark the box (the 'central rectangle') from -a to a horizontally and -b to b vertically; the asymptotes are the two diagonals of that box, extended. For the vertical hyperbola y^2/a^2 - x^2/b^2 = 1 the slopes are +-(a/b) instead.
The asymptotes capture the hyperbola's far-off behavior and its 'opening angle' in two simple lines, and they double as a drawing scaffold — sketch the box, draw the diagonals, then curve each branch from its vertex out toward the diagonals. They also reveal a special case: when a = b the asymptotes are perpendicular (slopes 1 and -1), giving a rectangular (equilateral) hyperbola, of which y = 1/x is a rotated example.
For x^2/16 - y^2/9 = 1, a = 4 and b = 3, so the asymptotes are y = (3/4)x and y = -(3/4)x. The central box runs from (-4, -3) to (4, 3); its diagonals are exactly these lines. At x = 100 the upper branch sits at y = (3/4)sqrt(100^2 - 16) = 74.994..., almost exactly on the line y = (3/4)(100) = 75.
The asymptotes are the diagonals of the a-by-b central box; the curve never touches them.
The hyperbola approaches its asymptotes but never meets or crosses them — that is the meaning of 'asymptote'. Parabolas and ellipses have no asymptotes; only the hyperbola, among the conics, runs off to infinity along straight guide lines.