Removable Discontinuity: A removable discontinuity is a point on the graph that is undefined or does not fit the rest of the graph. There is a gap at that location when you are looking at the graph. Rational functions are discontinuous when the denominator equals zero. To find the removable discontinuity:
How to Find:
1. Factor the numerator and the denominator
2. Identify factors that occur in both the numerator and the denominator
3. Set the common factors equal to zero
4. Solve for x
5. Write your answers in the form x=
6. To find t=y plug the x values in the remaining parts of the equation
Example:
(1)(has discontinuities at x=-2 and at x=3 so the denominator would be equal to zero.)
Example:
(2)Solve for x:
(3)Solve for y:
(4)Removable Discontinuity at
(5)Jump Discontinuity: If the left and right hand limits at x=a exist but disagree, then the graph jumps at x=a.
Example:The point x=2 is a Jump Discontinuity.
Infinite Discontinuity: Exists when one of the one-sided limits of the function is infinite.Example: The graph below shows a function that is discontinuous at x=a
Since the function never reaches a final value, the limit does not exist.