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.
congrats on reading the definition of vertical asymptote. now let's actually learn it.
Vertical asymptotes are found by setting the denominator of a rational function equal to zero and solving for $x$.
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.
Vertical asymptotes can be identified by evaluating one-sided limits; specifically, check $\lim_{{x \to a^+}} f(x)$ and $\lim_{{x \to a^-}} f(x)$.
A function can have more than one vertical asymptote, each corresponding to different values where the denominator is zero.
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?
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.