What It Means to Remove a Discontinuity
A function has a removable discontinuity when the function has a hole at a point, but the limit as the function approaches that point exists. In other words, the graph is missing a single point even though the function is trying to approach a perfectly well-defined value.
The key idea is simple: if the limit exists, we can usually remove the discontinuity by defining the function at that point to equal the limit.
This is one of the most important connections between limits and continuity in AP Calculus. The limit tells us what value the function should have at the point. We can then redefine the function so that it actually has that value.
How to Recognize a Removable Discontinuity
Suppose you are given a function that contains a factor in both the numerator and denominator. Substituting the troublesome value may produce , which tells you that direct substitution does not work.
But does not automatically mean the limit does not exist. It may simply mean that the original expression has a common factor that can be canceled.
For example, consider:
At , the original function is undefined. However, factoring the numerator gives us:
For values of other than 3, the expression simplifies to . That means the graph is the same as the line , except that the original function is missing the point where .
The Three-Step Method
- Find the problematic value. Determine where the original function is undefined or fails to be continuous.
- Find the limit. Factor, simplify, rationalize, or otherwise rewrite the function until you can evaluate the limit.
- Redefine the function. Set the function equal to the limit at the missing point.
For a removable discontinuity, the goal is to create a new function that satisfies the three conditions for continuity:
The important point is that changing the value at one point does not change the surrounding limit.
Worked Examples
Find the value that should be assigned to so that the function is continuous at .
Determine what value of makes the function continuous at .
Find the value that makes the following function continuous at .
Suppose
Find the value of that makes continuous at .
Why Cancellation Works
There is an important subtlety here. When you cancel a factor such as , you are not claiming that the original function is defined at the canceled value.
Instead, you are creating a simpler expression that has the same values everywhere except at the point where the original expression was undefined.
For example:
The two expressions agree whenever , but the original expression is undefined at . This is exactly why a hole appears in the graph.
The simplified expression tells us what the original function approaches as . We then use that limit to fill the hole.
Removable vs. Nonremovable Discontinuities
Not every discontinuity can be removed by simply redefining one point. The crucial question is whether the two-sided limit exists as a finite number.
If the left-hand and right-hand limits are different, there is a jump discontinuity. If the function approaches infinity or negative infinity, there is an infinite discontinuity. Neither can be fixed by changing the function's value at just one point.
Removable discontinuity
A hole occurs, but the function approaches the same finite value from both sides. One point can be redefined to repair the function.
Jump discontinuity
The left-hand and right-hand limits are different. No single value can make both sides agree.
Infinite discontinuity
The function grows without bound near the point. Assigning a finite value at the point cannot remove the discontinuity.
Common Mistakes
is an indeterminate form, not the final answer. It tells you that direct substitution has not yet revealed the limit. Look for factoring, cancellation, rationalization, or another algebraic technique.
If the denominator is zero at the troublesome point, the original function is undefined there. Instead, simplify the expression first and use the resulting expression to evaluate the limit.
Finding the limit is only half of the problem. To remove the discontinuity, define the function at that point to be exactly the value of the limit.
A hole can be repaired by redefining one point, but a jump or infinite discontinuity cannot. Always check whether the two-sided limit exists and is finite before trying to remove the discontinuity.
Practice Problems
Find the value of that makes the function continuous at .
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Find the value of that removes the discontinuity in .
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Find the value of that makes the function continuous at .
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Can the discontinuity in be removed by defining a value at ?
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