The graph of f(x)= (x^2-2x-8)/(x+6) has which of the following?

a. point discontinuity
b. vertical asymptote
c. both of the above
d. neither of the above

I'm stuck between b and c. I know that the vertical asymptote is x=-6. Help?

f(x)= (x^2-2x-8)/(x+6)

to have a point discontinuity, (x+6) should have been a factor of the numerator and cancelled out.
e.g. if g(x) = (x^2 - x - 12)/(x-4)
g(x) = (x-4)(x+3)/(x-4)
= x+3, x ≠ 4

There would have been a discontinuity at the point (4,7)

So in your question , we should just go with b)

Ohhh okay thank you for explaining

To determine whether the graph of the function f(x) = (x^2-2x-8)/(x+6) has a point discontinuity, a vertical asymptote, or both, we can analyze its behavior around x = -6.

First, let's find the vertical asymptote:
A vertical asymptote occurs when the denominator of the rational function is equal to zero, resulting in an undefined value. In this case, the denominator is x + 6. Setting it equal to zero:
x + 6 = 0
x = -6

So, the function has a vertical asymptote at x = -6.

Next, let's determine whether there is a point discontinuity at x = -6:
To check for a point discontinuity, we need to evaluate the limit of the function as x approaches -6 from both the left and the right.

Limit as x approaches -6 from the left-hand side:
(x^2-2x-8)/(x+6)
Substituting -6 for x:
((-6)^2-2(-6)-8)/(-6+6)
Simplifying:
(36+12-8)/0
The denominator is zero, resulting in an undefined value. Hence, we cannot determine the limit from the left-hand side.

Limit as x approaches -6 from the right-hand side:
(x^2-2x-8)/(x+6)
Substituting -6 for x:
((-6)^2-2(-6)-8)/(-6+6)
Simplifying:
(36+12-8)/0
Here again, the denominator is zero, resulting in an undefined value. Hence, we cannot determine the limit from the right-hand side.

Since both the left-hand and right-hand limits are undefined, it indicates that there is a point discontinuity at x = -6.

Therefore, the correct answer is c. The graph of the function f(x) = (x^2-2x-8)/(x+6) has both a point discontinuity and a vertical asymptote.