Question

Help please!!

For the following graph, what type of discontinuity occurs at x = 6 and what step of continuity is broken?

A) infinite and f(6) does not exist.
B) removable and f(6) does not exist.
C) jump and limx→6− f(x) ≠ limx→6+ f(x).
D) infinite and limx→6− f(x) ≠ limx→6+ f(x).

Answer 1

A) Infinite and f(6) does not exist

Explanation:

The three types of discontinuities are:

1. Removable or Hole: When the left hand limit is same as right hand limit but they are not equal to the value of function at x = 6.

Graphically if there exist a hole at the point x = 6, removable discontinuity occurs. But, it is not the case in the given graph.

2. Jump: When the left hand limit is not same as right hand limit.

But graphically, we see that both left and right limit at x=6 are tending to infinity i.e. both limits are same.

3. Infinite: It occurs when there exists vertical or horizontal asymptotes between the function and both limits tend to infinity.

Graphically, we see that the function has infinite discontinuity as there exist vertical asymptote and the value of function does not exist at x = 6.