AI Tutor
What happens when you divide a fixed number like 5 by a tiny number getting closer and closer to zero, such as 0.0010.001, 0.000010.00001, or 0.00000010.0000001?
The quotient explodes toward positive or negative infinity, creating an impassable barrier on a graph known as a vertical asymptote.
How can you find the exact equation of every vertical asymptote using algebra alone, without graphing a single point?

The Core Rule

A rational function is a fraction of two polynomials, written as f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}, where the bottom polynomial cannot be identically zero.
A vertical line x=cx = c is a vertical asymptote of f(x)f(x) if and only if cc makes the denominator zero (Q(c)=0Q(c) = 0) while leaving the numerator nonzero (P(c)β‰ 0P(c) \neq 0).
Let's see how this rule works when we analyze a rational function with quadratic polynomials.

Finding Asymptotes Step-by-Step

Consider the function f(x)=x+3x2βˆ’4f(x) = \frac{x + 3}{x^2 - 4}. First, factor both the top and bottom completely to reveal all candidate values.
The denominator factors as a difference of squares: f(x)=x+3(xβˆ’2)(x+2)f(x) = \frac{x + 3}{(x - 2)(x + 2)}, giving denominator zeros at x=2x = 2 and x=βˆ’2x = -2.
Next, evaluate the numerator P(x)=x+3P(x) = x + 3 at each candidate: P(2)=5β‰ 0P(2) = 5 \neq 0 and P(βˆ’2)=1β‰ 0P(-2) = 1 \neq 0. Because neither test equals zero, both lines x=2x = 2 and x=βˆ’2x = -2 are vertical asymptotes.