Polynomial and Rational Functions

Identify the vertical asymptote(s) of the rational function $f(x) = \frac{x+1}{x-2}$.

What is a vertical asymptote of a rational function?

Find the vertical asymptote(s) of the function $f(x) = \frac{x^2 - 1}{x^2 - 2x + 1}$.

Determine all vertical asymptotes of the function $f(x) = \frac{(x-3)(x+1)}{(x+1)(x-2)^3}$.

Which of the following functions has a hole at $x = 1$?

Given the function $f(x) = \frac{x-4}{(x-4)(x+2)}$, determine whether $x=4$ corresponds to a vertical asymptote or a hole.

Given the function $f(x) = \frac{(x-1)(x+2)}{(x-1)(x-3)}$, describe the behavior of the function near $x=1$.

For the rational function $r(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions, vertical asymptotes often occur where:

What is the vertical asymptote(s) of the function $f(x) = \frac{x}{x^2 - 4}$?

Determine the vertical asymptote(s) of the function $f(x) = \frac{x+3}{x^2 + 5x + 6}$.