What kind of discontinuity is this:
F(x)= (2x^2 - 5x -3)/(x-3) if x does not equal 3
6 if x=3
It is a pieces use function.
I thought it was removable but my answer key says it is a jump discontinuity. This is from an exam review and I want to get it right.
I meant to say it is a piecewise function
everywhere except x=3, F(x) = 2x+1
Since 6≠7 you have a jump there. The limit from both sides is 7, but F is defined to be 6, and there's no way to close that gap.
To determine the type of discontinuity of the given function, let's first analyze the behavior of the function as x approaches the point of interest, which in this case is x = 3.
For values of x not equal to 3, the function is defined as F(x) = (2x^2 - 5x - 3)/(x - 3). We can simplify this expression:
F(x) = (2x + 1)(x - 3)/(x - 3)
Here, we can cancel out the common factor of (x - 3) in the numerator and denominator, leaving us with:
F(x) = 2x + 1
Now, if we substitute x = 3 into the original function, we get F(3) = 6. So at x = 3, the value of the function is 6.
To determine the type of discontinuity, we compare the limit of the function as x approaches 3 from the left and right sides. Let's evaluate these limits:
Limit as x approaches 3 from the left-hand side (x < 3):
lim(x->3-) (2x + 1) = 2 * 3 + 1 = 7
Limit as x approaches 3 from the right-hand side (x > 3):
lim(x->3+) (2x + 1) = 2 * 3 + 1 = 7
Both limits are equal to 7, which means the function approaches the same value from both directions as x approaches 3.
In this case, since the function has a defined value (6) at x = 3 and the limits from both sides approach the same value (7), we have a jump discontinuity.
Therefore, your answer key is correct, and the type of discontinuity for the given function is a jump discontinuity at x = 3.