What Makes Piecewise Limits Different?
A piecewise function uses different formulas for different values of . That means you cannot always evaluate a limit by looking at just one formula. You first have to determine which branch of the function applies on each side of the point you are approaching.
For example, consider the piecewise function:
Suppose we want to find the limit as . Values of that approach 2 from the left use the first branch, while values approaching 2 from the right use the second branch.
This is the key idea: for a piecewise function, always look at the side from which you are approaching the boundary point.
The Key Rule for Two-Sided Limits
A two-sided limit exists only when the left-hand and right-hand limits both exist and are equal.
For a piecewise function, this becomes especially important at a point where the formula changes. You typically need to calculate two separate limits and compare them.
A Step-by-Step Method
- Find the point you are approaching. Identify the value in the limit.
- Look at the piecewise breakpoints. Determine whether the function changes formulas at that value.
- For the left-hand limit, use the branch defined to the left of the point.
- For the right-hand limit, use the branch defined to the right of the point.
- Evaluate both limits. Direct substitution usually works once the correct branch has been selected.
- Compare the answers. If they are equal, the two-sided limit exists. If they differ, the two-sided limit does not exist.
Worked Examples
Evaluate where
Evaluate where
Consider
Find .
Suppose
Find the value of that makes the limit exist.
How to Read the Inequalities
The inequalities attached to each branch tell you exactly when that formula should be used.
If you are approaching from the left, your nearby values satisfy , so you use the first formula. From the right, nearby values satisfy , so you use the second formula.
It does not matter which branch contains the actual endpoint when you are calculating a limit. What matters is the behavior of the function for values near the point.
Limits at a Point Where the Branches Meet
Piecewise functions often use different formulas on either side of a boundary. A common AP Calculus question asks whether the resulting function is continuous at that boundary.
For continuity at , three conditions must hold:
- must be defined.
- must exist.
- .
For a piecewise function, the second condition is where the one-sided limits become especially important. If the left and right limits disagree, the two-sided limit does not exist, and the function cannot be continuous there.
Common Mistakes
The most common error is substituting the approaching value into whichever formula appears first. Instead, ask whether you are approaching from the left or right and select the branch accordingly.
The value of at the point does not determine the limit. A limit describes what happens as gets close to the point, not necessarily what happens exactly at the point.
Always compare the left-hand and right-hand limits at a piecewise breakpoint. If they are different, the correct answer is that the two-sided limit does not exist.
When a branch says and another says , pay attention to which branch contains the endpoint. For a limit, however, the endpoint's inclusion does not change the nearby behavior.
Practice Problems
Evaluate where .
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Evaluate where .
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Does exist if ?
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Find so that exists, where .
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