Determine whether the function
is continuous at each given value. If not, explain.
- x = -5
- x = -4
- x = 0
- x = 2
- x = 3
Answer
- When we try to evaluate f(-5), we find which is undefined. Since f(-5) does not exist, f is discontinuous at x = -5.
- When we try to evaluate f(-4), we find f(-4) = 0, which is fine. We will see if exists and agrees with f(-4). As x approaches -4 from the left, we use the part of the function definition that says The left-sided limit is When x is approaches -4 from the right, x is just a bit bigger than -4, we use the part of the function definition that says f(x) = x - 8 for -4 < x ≤ 2. Then the right-sided limit is Since the left- and right-hand limits agree, Alas, since f(-4) = 0, the limit disagrees with the function value, therefore f is discontinuous at x = -4.
- f(0) = 0 - 8 = -8. As x approaches 0 from the left or the right, we use the part of the function definition that says f(x) = x-8 for -4the function is continuous at x = 0.
- f(2) = 2 - 8 = -6, which is fine. Now we will check the limit. As x approaches 2 from the left we use the part of thefunction definition that says f(x) = x-8 for -4 < x ≤ 2 we find As x approaches 2 from the right we use the part of the function definition that says and from earlier work with limits we know that is undefined. Therefore does not exist, therefore f is discontinuous at x = 2.
- f(3) we findwhich is undefined. Therefore f is discontinuous at x = 3.