Functions, Graphs, and Limits
Vertical asymptotes most frequently show up in rational functions. When a rational function f(x) has a non-zero constant in the numerator and an expression with a variable in the denominator, the function f(x) will have vertical asymptotes at all values of x that make the denominator 0. If the denominator has no roots, then f(x) will have no vertical asymptotes.
This works for the same reason that 1 / x has a vertical asymptote at zero: the numerator is a non-zero constant and the denominator is getting smaller and smaller, therefore the fraction will get bigger and bigger.
If we use a different constant, the principle is still the same.
If a rational function has something besides a non-zero constant in the numerator, we may need to be creative: factor the numerator and denominator, cancel if possible, and find the roots of whatever polynomial remains in the denominator after factoring.
Be Careful: Remember to simplify the rational functions!