Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the input's domain. Finding the limit of a piecewise function at a point requires checking the limits from the left and right. If both one-sided limits exist and are equal, then the limit at that point exists and is equal to that common value. If the one-sided limits are not equal, the limit does not exist.
For High School Calculus, understanding these limits is crucial for continuity and differentiability. The worksheet below provides practice in evaluating limits of piecewise functions.
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Limit | A. A function defined by multiple sub-functions. |
| 2. Piecewise Function | B. The value that a function approaches as the input approaches some value. |
| 3. One-Sided Limit | C. The limit of a function as the input approaches a value from either the left or the right. |
| 4. Continuity | D. The property of a function such that a small change in the input results in a small change in the output. |
| 5. Discontinuity | E. A point where the function is not continuous. |
A piecewise function is defined by _______ sub-functions. To find the _______ at a point, check the _______ from the left and right. If these are _______, the limit exists.
Explain, in your own words, why it is important to check both the left-hand and right-hand limits when evaluating the limit of a piecewise function at a point where the function definition changes. Give an example to illustrate your explanation.
Please log in to post your answer.
Log InEarn 2 Points for answering. If your answer is selected as the best, you'll get +20 Points! 🚀