Finding Vertical Asymptotes - At A Glance

Let . Find all vertical asymptotes (if any) of f(x).

Since f(x) has a constant in the numerator, we need to find the roots of the denominator. The denominator factors as x² – 5x + 6 = (x – 2)(x – 3), therefore the roots are x = 2 and x = 3. These are also the vertical asymptotes.

Let . Find all vertical asymptotes (if any) of f(x).

Again, we need to find the roots of the denominator. Since x² + 1 is never zero, there are no roots. This function has no vertical asymptotes.

Let Find the vertical asymptote(s) of f, if there are any.

We can factor:

Since now we have a constant in the numerator and x in the denominator, this function has a vertical asymptote at x = -2.

Let f(x) = 5x² + 3. Find the vertical asymptote(s) of f, if there are any.

This is a polynomial. Since f(x) is defined for every value of x, and there are no finite values of a (that is, actual numbers a) near which
f(x) keeps getting bigger and bigger and bigger, this function has no vertical asymptotes.

If a rational function doesn't have a constant in the numerator, we do the same stuff as before: factor the numerator and denominator, simplify if possible, and find the roots of the denominator. The roots of the simplified denominator are the vertical asymptotes.

If the value x = a only makes the denominator zero, the numerator acts like a finite number as x approaches a. Meanwhile the denominator
gets crazy small, making the value of the function get crazy big.

Find the vertical asymptotes of

First we simplify the function:

As x approaches 3, the numerator, x – 2, approaches 1. When x is close to 3, the function f(x) is pretending to be the function , which we know has a vertical asymptote at x = 3. Therefore our original function, f(x), has a vertical asymptote at x = 3 too.