Why do the simplified functions agree with the original expressions near the target point even when the originals are undefined there?
Conditions
- The target point is excluded from the domain of the original expression.
- The algebraic simplification is valid for all near the target point ( target).
- The limit exists for the simplified function.
Reasoning, step by step
- Recognize that limits concern the behavior of inputs approaching a point, not evaluation at the point.
- Verify that the simplification steps (e.g., cancelling ) are valid for .
- Conclude that the simplified function and original function are identical in the neighborhood of the limit point.
- Apply the principle that identical functions in a punctured neighborhood have the same limit.
Example
The summary states: "The simplified functions agree near the target point even when the original expressions are undefined there." The script for the first example notes: "Cancellation concerns nearby inputs, not evaluation of the original expression at its missing point."
Common misconceptions
- Believing that if a function is undefined at a point, its limit cannot exist.
- Thinking that algebraic simplification changes the function's behavior everywhere, rather than just removing the discontinuity at the specific point.
- Confusing the value of the function at the point with the limit of the function as it approaches the point.
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