Vertical Asymptote of a Rational Function

A vertical asymptote (VA) of a function is an imaginary vertical line to which its graph appears to be very close but never touch. It is of the form x = some number. Here, “some number” is closely connected to the excluded values from the domain. But note that there cannot be a vertical asymptote at x = some number if there is a hole at the same number. A rational function may have one or more vertical asymptotes. So to find the vertical asymptotes of a rational function:

• Simplify the function first to cancel all common factors (if any).
• Set the denominator = 0 and solve for (x) (or equivalently just get the excluded values from the domain by avoiding the holes).

Example: Find the vertical asymptotes of the function f(x) = (x² + 5x + 6) / (x² + x – 2).

Solution:

We have already seen that this function simplifies to f(x) = (x + 3) / (x – 1).

Setting the denominator to 0, we get

x – 1 = 0
x = 1

Thus, there is a VA of the given rational function is, x = 1.