Hole at x=0 and 1 other point of discontinuity.
A hole in a graph occurs when a point is missing in the graph due to a removable discontinuity
A hole in a graph occurs when a point is missing in the graph due to a removable discontinuity. This typically happens when there is a factor that can be canceled out in both the numerator and denominator of a rational function. The canceled factor creates a hole at that specific x-value.
On the other hand, a point of discontinuity refers to a non-removable discontinuity where the function doesn’t exist or approaches different values from different directions.
Now, if you have a hole at x = 0 and one other point of discontinuity, it means that your function has a missing point at x = 0 while having another type of discontinuity at a different x-value.
To clarify this further, let’s consider an example:
Suppose we have the function f(x) = (x-1)/(x-1), which simplifies to f(x) = 1 for all x ≠ 1.
In this case, we have a removable discontinuity or a hole at x = 1. This means that the graph of the function would have a hole at the point (1, 1) since the factor (x-1) can be canceled out.
Now, let’s say we have another point of discontinuity at x = 2. For example, suppose we have g(x) = 1/(x-2).
In this case, there is a non-removable discontinuity at x = 2. The graph of the function would have a vertical asymptote at x = 2, indicating that the function doesn’t exist at x = 2.
To summarize, if you have a hole at x = 0 and one other point of discontinuity, it means that the graph of the function would have a missing point at x = 0 due to a removable discontinuity (hole) and some other type of non-removable discontinuity at a different x-value.
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