In AP Calculus, limits describe the behavior of a function as its input approaches a certain value. Continuity, on the other hand, means that a function has no breaks or jumps; you can draw it without lifting your pen. A function is continuous at a point if the limit exists at that point, the function is defined at that point, and the limit equals the function's value. Understanding these concepts is crucial for mastering derivatives and integrals!
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Limit | A. A function where you can draw without lifting your pen |
| 2. Continuity | B. A value that a function approaches as the input approaches some value |
| 3. Removable Discontinuity | C. A point where the limit exists, but the function value doesn't match, or isn't defined. |
| 4. Infinite Discontinuity | D. A point where the function approaches infinity (vertical asymptote). |
| 5. Jump Discontinuity | E. A point where the function 'jumps' from one value to another. |
A function $f(x)$ is said to be __________ at a point $x = a$ if the $\lim_{x \to a} f(x)$ __________, $f(a)$ is __________, and $\lim_{x \to a} f(x) = f(a)$. If any of these conditions are not met, the function is __________ at $x = a$.
Explain, in your own words, why understanding limits is essential for understanding the concept of a derivative. Give a real-world example of something that can be modeled using limits and continuity.
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