Question

Which of the following discontinuity is found by reducing common factors in the numerator and denominator of a rational function?
a. Vertical Asymptote
b. Horizontal Asymptote
c. Jump Discontinuity
d. Hole​

Answer 1

We can define discontinuity as a point in our function where we have a given indetermination. Here particularly we refer to the indetermination of dividing by zero.

The correct option is a: Vertical Asymptote

Here we work with functions like:

$f(x) = \frac{g(x)}{k(x)}$

Suppose that g(x) = (x + 1)*(x - 2)

                        k(x) = (x + 1)*(x + 17)

Then we have:

$f(x) = \frac{(x + 1)*(x - 2)}{(x + 1)*(x + 17)}$

Here we have the common factor (x + 1) that we can remove, doing that we get:

$f(x) = \frac{(x - 2)}{(x + 17)}$

This function can't be reduced any more, so now we can see the discontinuity. It happens when the denominator is equal to zero at x = -17.

Now, if any number is divided by a really smaller number, then the outcome of the quotient is really large (or really small, depending of the sign). From this, for dividing by zero we will have vertical asymptotes that tend to infinity (or negative infinity).

Then the discontinuity that is found by reducing common factors in the numerator and denominator of a rational function is a vertical asymptote, thus the correct option is a.

If you want to learn more, you can read:

brainly.com/question/17317969