Finding Horizontal and Vertical Asymptotes of Rational Functions

An asymptote is a horizontal/vertical oblique line whose distance from the graph of a function keeps decreasing and approaches zero, but never gets there.

In this wiki, we will see how to determine horizontal and vertical asymptotes in the specific case of rational functions. (Functions written as fractions where the numerator and denominator are both polynomials, like \( f(x)=\frac{2x}{3x+1}.)\)

For horizontal asymptotes in rational functions, the value of \(x\) in a function is either very large or very small; this means that the terms with largest exponent in the numerator and denominator are the ones that matter. For example, with \( f(x) = \frac{3x^2 + 2x - 1}{4x^2 + 3x - 2} ,\) we only need to consider \( \frac{3x^2}{4x^2} .\) Since the \( x^2 \) terms now can cancel, we are left with \( \frac{3}{4} ,\) which is in fact where the horizontal asymptote of the rational function is.

In a case like \( \frac{3x}{4x^3} = \frac{3}{4x^2} \) where there is only an \(x\) term left in the denominator after the reduction process above, the horizontal asymptote is at 0.

In a case like \( \frac{4x^3}{3x} = \frac{4x^2}{3} \) where there is only an \(x\) term left in the numerator after the reduction process above, there is no horizontal asymptote at all. (There may be an oblique or "slant" asymptote or something related.)

A vertical asymptote with a rational function occurs when there is division by zero. For example, with \( f(x) = \frac{3x}{2x -1} ,\) the denominator of \( 2x-1 \) is 0 when \( x = \frac{1}{2} ,\) so the function has a vertical asymptote at \( \frac{1}{2} .\)

Find the vertical asymptote of the graph of the function

\[f(x)=\dfrac{3}{x-2}.\]

The denominator \( x - 2 = 0 \) when \( x = 2 .\) Thus the line \(x=2\) is the vertical asymptote of the given function. \(_\square\)

Graph of the function

Find the vertical asymptotes of the graph of the function

\[f(x)=\dfrac{4}{x^2-25}.\]

\( x^2 - 25 = 0 \) when \( x^2 = 25 ,\) that is, when \( x = 5 \) and \( x = -5 .\) Thus this is where the vertical asymptotes are.

Verifying the obtained Asymptote with the help of a graph