Which Of The Functions Graphed Below Has A Removable Discontinuity

Identifying Removable Discontinuities: A Visual and Algebraic Guide

Understanding discontinuities in functions is a cornerstone of calculus and advanced algebra, bridging the gap between abstract equations and their tangible graphical representations. Among the various types of discontinuities—jump, infinite, and essential—the removable discontinuity holds a unique and often misunderstood place. So it appears as a single "hole" in an otherwise smooth curve, a point where the function is undefined but could easily be "fixed. " This article provides a comprehensive, visual, and algebraic roadmap to identifying this specific flaw in a graph, equipping you with the analytical tools to confidently pinpoint it, even without an explicit equation.

What Exactly is a Removable Discontinuity?

A removable discontinuity occurs at a point x = a in the domain of a function where the function fails to be defined or fails to match its limiting value, yet the limit as x approaches a exists and is finite. The discontinuity is "removable" because we could define a new function that assigns f(a) to be equal to the limit L, thereby "plugging" the hole and making the function continuous at that point. Practically speaking, graphically, this manifests as a hole—an open circle on the curve. Day to day, the key characteristic is that the "break" is isolated to that single point. The function behaves perfectly normally on either side of a; the issue is solely at a itself. Think of it as a single missing pixel in an otherwise perfect image; the image's structure is sound, but one data point is absent.

The Two-Pronged Identification Strategy: Graphical and Algebraic

To determine which graphed function has a removable discontinuity, you must employ a dual approach: visual inspection for the telltale hole, followed by algebraic verification to confirm the limit exists. Relying on sight alone can be deceptive, as other discontinuities can sometimes mimic a hole at a glance.

Step 1: The Graphical "Hole" Test

Carefully examine each graph. Look for these specific visual cues:

• An open circle (⦺) on the curve. Day to day, this is the most definitive sign. The curve approaches a specific (x, y) coordinate from both sides, but at that exact x-value, there is no filled-in point. Also, * A gap in a continuous-looking line or curve. The line should appear unbroken and smooth as it nears the gap from the left and the right, heading toward the same y-value.
• Contrast with other discontinuities:
  • A jump discontinuity shows the curve ending at two different y-values from the left and right (a sudden vertical step). Because of that, * An infinite discontinuity features a vertical asymptote, where the curve shoots up or down toward infinity near a specific x-value. * The hole is the only major discontinuity where the curve does not exhibit violent behavior (jumping or flying to infinity) near the problem point.

Step 2: The Algebraic Confirmation (The Limit Test)

Once a candidate hole is spotted visually, you must confirm it algebraically. ** Simplify the function algebraically (factor and cancel common terms) and then substitute x = a into the simplified expression. But **Calculate the limit as x approaches a. 3. The verdict: If lim (x→a) f(x) = L exists (a finite number) and f(a) is either undefined or f(a) ≠ L, then x = a is a removable discontinuity. Because of that, 2. And **Check if f(a) is undefined. ** This is often due to a factor in the denominator that cancels with the numerator. For a point x = a:

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1. The value L is the y-coordinate of the hole.

Worked Examples: From Graph to Conclusion

Since no specific graphs were provided, let's analyze common function types you are likely to encounter.

Example 1: The Rational Function with a Common Factor Consider f(x) = (x² - 4) / (x - 2).

• Graphically: This function graphs as the line y = x + 2, except at x = 2. At x = 2, there is a hole. The line is perfectly straight and continuous everywhere else.
• Algebraically: f(2) is undefined (division by zero). Factor: (x-2)(x+2)/(x-2). For x ≠ 2, this simplifies to x + 2. lim (x→2) f(x) = 2 + 2 = 4. Since the limit exists (4) and f(2) is undefined, there is a removable discontinuity at (2, 4).

Example 2: A Piecewise Function with a Mismatched Point g(x) = { x² for x ≠ 3; 5 for x = 3 }.

• Graphically: The parabola y = x² is drawn continuously. At x=3, the point on the parabola would be (3,9), but instead, a single isolated point is plotted at (3,5). The curve approaches (3,9) from both sides, but the function's value at 3 is defined as 5, not 9. This creates a hole at (3,9) because the limit is 9, but g(3)=5.
• Algebraically: lim (x→3) g(x) = 3² = 9. g(3) = 5. Limit exists, but g(3) does not equal the limit. Removable discontinuity at (3,9).

Example 3: The Non-Removable Discontinuity for Comparison h(x) = 1/(x-1).

• Graphically: A vertical asymptote at x=1. The curve on the left goes to negative infinity, and on the right to positive infinity. This is an infinite discontinuity, not removable. The limit does not exist (it's infinite).

Common Pitfalls and How to Avoid Them

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