Rational functions and asymptotes

Idea

A rational function is a quotient of two polynomials:

R (x) = \frac{P ( x )}{Q ( x )}

$Q$ is not identically zero. To read asymptotes and discontinuities, start from the cancelled quotient $\frac{P ( x )}{Q ( x )}$ .

Vertical asymptote: after common factors are cancelled, if $x = a$ makes the denominator zero and the numerator nonzero, then $x = a$ is a vertical asymptote. As $x \to a^{\pm}$ , $R (x)$ may go to $+ \infty$ or to $- \infty$ , and the graph is unbounded in the vertical direction
Removable discontinuity (hole): if the numerator and the denominator are both zero at $x = a$ , cancellation may leave only a hole. For example, $\frac{( x - 1 ) ( x + 2 )}{x - 1} = x + 2$ for $x \neq = 1$ : the main graph is the line $y = x + 2$ , with a hole at $x = 1$
Horizontal asymptote: read it from the leading terms after cancellation. If $de g P < de g Q$ , the graph approaches $0$ far away. If $de g P = de g Q$ , it approaches the ratio of the leading coefficients
Oblique asymptote: when $de g P = de g Q + 1$ after cancellation, division gives a linear quotient $S (x)$ . This page calls that far line an oblique asymptote, and $R (x) - S (x) \to 0$

Discontinuities, zeros, and asymptotes together are the skeleton of the graph.

Factors: build $R (x) = A \frac{x - r}{x - a}$ from Scale A, Zero r, and Asymptote a, and watch the graph, the zero, and the vertical asymptote update. The degrees match, so the horizontal asymptote is $y = A$
Hole: give the numerator and the denominator a common factor. Hole h places it, and Hole marks shows the hole left after cancellation. The drawn graph is $R (x) = A (x - r) (x - h) / [(x - a) (x - h)]$ , with a vertical asymptote at $x = a$ and a horizontal asymptote $y = A$
Horizontal: compare the degrees after cancellation and watch the far height. This control draws $R (x) = b + A / (x - a)$ . Shift b is that height when the degrees match. The term $A / (x - a)$ is the remainder, of lower degree. If $b = 0$ , the numerator degree drops and the line is $y = 0$
Oblique asymptote: when the degree gap is $1$ , show the linear quotient. Slope m, Shift b, Remainder c, and Asymptote a draw $R (x) = m x + b + c / (x - a)$ , and $S (x) = m x + b$ is the far line
Asymptotes and Hole marks: show or hide those guides. Advanced opens the line-by-line cancellation and splitting

Suggested order: Factors, Hole, Horizontal, then Oblique asymptote.

A blow-up or a hole nearby, and a horizontal or oblique line far away, both depend on what remains after cancellation. The same quotient reads as a different skeleton at a different scale.
A denominator zero is not always a vertical wall. After it is cancelled, only a hole may remain. The near reading and the far reading use that same cancelled quotient.
Horizontal and oblique are two far skeletons. They do not replace a discontinuity nearby. They read the same $R (x)$ once more at infinity.