Vertical asymptote

Definition

A vertical asymptote is a line $x = a$ where the function $f(x)$ approaches positive or negative infinity as $x$ approaches $a$. Vertical asymptotes occur at values of $x$ that make the denominator of a rational function zero, provided that the numerator does not also become zero at those points.

5 Must Know Facts For Your Next Test

1. Vertical asymptotes are found by setting the denominator of a rational function equal to zero and solving for $x$.
2. If both the numerator and the denominator approach zero simultaneously, further analysis using limits is required to determine if there is a vertical asymptote or a hole.
3. Vertical asymptotes can be identified by evaluating one-sided limits; specifically, check $\lim_{{x \to a^+}} f(x)$ and $\lim_{{x \to a^-}} f(x)$.
4. A function can have more than one vertical asymptote, each corresponding to different values where the denominator is zero.
5. The presence of a vertical asymptote indicates an infinite discontinuity in the graph of the function.

Review Questions

• How do you determine where vertical asymptotes occur in a given rational function?
• Why must you consider one-sided limits when analyzing vertical asymptotes?
• What is the difference between a hole and a vertical asymptote in terms of their impact on the graph?

Related terms

Limit:

The value that a function approaches as the input approaches some value.

Rational Function:

A function that can be expressed as the ratio of two polynomials.

Hole (Removable Discontinuity): A point on the graph where a function is not defined due to both numerator and denominator being zero, but which could be 'filled' by redefining the function at that point.