Rational Functions and Asymptotes

Allowing x into the denominator creates lines the curve approaches but never reaches - a vertical asymptote at every root of the denominator, and at most one horizontal one decided by comparing degrees.

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In a polynomial, only ever appears on top, so nothing unusual can happen. A rational function allows in the denominator too — and that one change produces the lines a curve approaches but never reaches.

Concept What a rational function is
One polynomial divided by another, with the denominator never allowed to be zero.

Ordinary polynomials are the special case , and constants the case where is a number as well. Both are rational functions — they simply never run into the difficulty, because their denominator can never vanish.

Theorem Vertical asymptotes: where the denominator is zero
x = 0 y = 0 1/x
If , then is a vertical asymptote. The curve shoots off towards infinity there, coming ever closer to the line without ever meeting it.
— denominator zero at , so one vertical asymptote
— zero at and , so two

The count is simply the number of roots of the denominator, so there is no limit to how many a function may have.

Theorem Horizontal asymptotes: compare the degrees

A horizontal asymptote describes what happens far out to the left and right. Which case applies depends only on the two degrees:

Denominator degree higher →
Degrees equal → , the ratio of the leading coefficients
Numerator degree higher → none at all

Unlike vertical asymptotes, there is at most one horizontal asymptote — never two. The three cases above are exhaustive, so checking the degrees settles the question immediately.

Example Both asymptotes for 1/x
Step 1 — the denominator is , zero at ⟹ vertical asymptote
Step 2 — degrees: numerator 0, denominator 1
Step 3 — the denominator wins ⟹ horizontal asymptote
⟹ the curve hugs both axes without ever touching them

This is the shape drawn above: two separate branches, each squeezed between the two dashed lines.

Example Equal degrees

Find the horizontal asymptote of .

Step 1 — degrees: numerator 2, denominator 2 — equal
Step 2 — take the leading coefficients: 2 on top, 3 below
⟹ horizontal asymptote

Only the leading terms matter here. Far from the origin and dwarf everything else, so the fraction settles towards .

Summary
  1. A rational function is P(x) ÷ Q(x), with Q(x) never zero.
  2. Polynomials and constants are special cases where the denominator cannot vanish.
  3. Every root of the denominator gives a vertical asymptote, so there may be many.
  4. There is at most one horizontal asymptote, decided by comparing degrees.
  5. Denominator higher → y = 0; equal → y = ratio of leading coefficients; numerator higher → none.
  6. For 1/x both axes are asymptotes; for (2x²+1)/(3x²+x) the horizontal asymptote is y = 2/3.
283. Relations and Functions
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