if f is a function that has a removable discontinuity at x = 3, which of the following could be the graph of…
if f is a function that has a removable discontinuity at x = 3, which of the following could be the graph of f?
Answer
Explanation:
Step1: Recall definition of removable discontinuity
A removable discontinuity at $x = a$ occurs when $\lim_{x\rightarrow a}f(x)$ exists, but $f(a)$ is either undefined or not equal to the limit. Graph - ically, it appears as a hole in the graph.
Step2: Analyze Option A
In Option A, as $x$ approaches 3 from the left and from the right, the function values are approaching different values. This is a jump - discontinuity.
Step3: Analyze Option B
In Option B, as $x$ approaches 3 from the left and from the right, the function values are approaching the same value. There is a hole at $x = 3$ (open - circle) and a non - matching point (closed - circle) at $x=3$, which is characteristic of a removable discontinuity.
Answer:
B.