Rational Functions and Holes

#1.10 Rational Functions and Holes

#🏌🏽 Hole in One!

If the multiplicity of the zero 'a' in the numerator is greater than or equal to its multiplicity in the denominator, then the function has a hole at x = a. Remember, multiplicity is the number of times a zero appears in the factorization. For example, if 'a' appears 3 times in the numerator and 2 times in the denominator, there's a hole at x = a.

To find the y-value of the hole, cancel the common factor and then evaluate the simplified function at x = a. Let's look at an example:

r(x) = (x² - 9) / (x - 3)

The numerator has zeros at x = -3 and x = 3, while the denominator has a zero at x = 3. The zero at x=3 has a multiplicity of 2 in the numerator and 1 in the denominator, so there is a hole at x=3. First, factor the numerator:

r(x) = (x + 3)(x - 3) / (x - 3)

Then, cancel the common factor (x - 3):

r(x) = x + 3

Finally, plug in x = 3 to find the y-value of the hole:

r(3) = 3 + 3 = 6

So, there's a hole at (3, 6). 🤓

Graph displaying a hole at the intersection (1,1)

Source: Jed Q

#🌉 Connecting to Limits

When a rational function, r(x), has a hole at x = c, the function approaches a finite limit as x approaches c. This limit, L, is the y-coordinate of t...