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Question 8 • Mathematics III

Which of the graphed functions has a removable discontinuity?

Answer

Image with description On a coordinate plane, the scale on the x axis range from negative 8 to 8 in increments of 1, and the scale on the y axis range from negative 5 to 5 in increments of 1. The graph is comprised of 3 lines. The curve starts at the closed circle (0 comma 0) and curves up through (negative 2 comma 4). The first line starts at the closed circle (negative 1 comma negative 1) and goes through (3 comma negative 1). The second line starts at the closed circle (3 comma negative 1) and goes through (6 comma 3). Both lines are connected to each other. The third line starts at the open circle (negative 1 comma negative 2) and goes through (negative 5 comma negative 2).

Image with description On a coordinate plane, the scale on the x axis range from negative 8 to 8 in increments of 1, and the scale on the y axis range from negative 5 to 5 in increments of 1. The graph is comprised of a curve and 2 lines. The curve starts at the closed circle (0 comma 0) and curves up through (negative 2 comma 4). The first line starts at the closed circle (0 comma 0) and goes to (3 comma negative 3). The other line starts at the closed circle (3 comma negative 3) and goes through (5 comma 1). The curve and both the lines are connected to each other.

Image with description On a coordinate plane, the scale on the x axis range from negative 8 to 8 in increments of 1, and the scale on the y axis range from negative 5 to 5 in increments of 1. The graph is comprised of 4 lines. The first line starts at the open circle (3 comma negative 1) and goes through (negative 1 comma negative 1). The second line starts at (negative 1 comma negative 1) and goes through (negative 4 comma negative 3). Both lines are connected to each other. The third line starts at the open circle (5 comma negative 1) and goes through (5 comma negative 1). Two points are marked on the coordinate plane at (3 comma 2) and (5 comma 2) and they are joined by a line.

Explanation

A removable discontinuity occurs when the graph has a hole but approaches the same y-value from both sides. In choice C, both curves approach (0, 0), so the two-sided limit as x approaches 0 is 0, even though the point is open. The other choices show connected graphs or mismatched values from the two sides, rather than a hole with a matching limit.

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