Rational Functions: Asymptotes, Holes and Algebraic Fractions

Q: Can a graph cross its horizontal asymptote?

Yes, it can cross it for smaller values of x. A horizontal asymptote only describes what happens as x becomes very large or very negative.

How to simplify algebraic fractions by factorising and cancelling, how to find vertical and horizontal asymptotes, why some zeros of the denominator make holes instead, and how all of this is limits in disguise.

3 min read Supports 4 lessons

Divide one polynomial by another and you get a rational function, named after [[ratios|ratio]]. Its graph can do three things a polynomial's never can: shoot off to infinity, level out towards a line, or have a single point missing.

A vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 1$ .

Simplifying algebraic fractions

Algebraic fractions simplify exactly like number fractions: factorise the top and bottom, then cancel common factors.

$\frac{x^2 - 9}{x^2 + 5x + 6}$
Factorise top and bottom.

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Watch out

Only cancel factors, never terms. $\frac{x + 3}{x}$ is not $3$ .

Vertical asymptotes

Near a value where the denominator is zero (after cancelling), the denominator is tiny, so the fraction is huge. The graph rockets up or down alongside a vertical line called a vertical asymptote.

Vertical asymptotes

q(a) = 0 \text{ after cancelling} \;\Rightarrow\; x = a

Holes

If a factor cancels, the function is still undefined there, but the graph does not blow up. It just has a single missing point, a hole.

Worked example

Find the hole and vertical asymptote of $f(x) = \dfrac{x - 3}{x^2 - x - 6}$ .

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Factorise the denominator.
$\frac{x - 3}{(x - 3)(x + 2)}$

Horizontal asymptotes

For very large $x$ , only the highest-power terms matter. Compare the degrees of the top and bottom.

Horizontal asymptotes

\frac{3x^2 + 1}{x^2 - 5} \to 3 \;\text{ as }\; x \to \pm\infty

Same degree: the ratio of leading coefficients. Top degree lower: $y = 0$ . Top degree higher: no horizontal asymptote.

Tip

Try a big number. At $x = 1000$ , $\frac{3x^2 + 1}{x^2 - 5}$ is $\frac{3\,000\,001}{999\,995} \approx 3.00002$ .

Combining algebraic fractions

Adding and subtracting needs a common denominator, just as with numbers:

\frac{1}{x} - \frac{1}{x + h} = \frac{(x + h) - x}{x(x + h)} = \frac{h}{x(x + h)}

Key idea

Asymptotes are limits in disguise: $f(x) \to \pm\infty$ as $x \to 2$ , and $\lim_{x \to \infty} f(x) = 1$ . A hole is exactly the kind of point a limit can fill in.

Common questions

What is the difference between a hole and a vertical asymptote?: Both come from zeros of the denominator. If the factor cancels with one on top, the graph has a hole. If it does not cancel, the graph has a vertical asymptote.
Can a graph cross its horizontal asymptote?: Yes, it can cross it for smaller values of $x$ . A horizontal asymptote only describes what happens as $x$ becomes very large or very negative.

Practise rational functions

The Rational Functions & Asymptotes lesson has worked examples and unlimited practice that adapts to you.

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Common questions

What is the difference between a hole and a vertical asymptote?

Both come from zeros of the denominator. If the factor cancels with one on top, the graph has a hole. If it does not cancel, the graph has a vertical asymptote.

Can a graph cross its horizontal asymptote?

Yes, it can cross it for smaller values of $x$ . A horizontal asymptote only describes what happens as $x$ becomes very large or very negative.