📚 Topic Summary
One-sided limits focus on the value a function approaches as the input ($x$) gets closer to a specific point from either the left or the right. Unlike two-sided limits, which require the function to approach the same value from both directions, one-sided limits only consider one direction. Graphically, this means we're only looking at the function's behavior on one side of a particular $x$-value. Evaluating one-sided limits often involves analyzing graphs or applying limit properties tailored to the specific direction of approach.
Understanding one-sided limits is crucial because it helps us analyze functions with discontinuities or piecewise definitions where the function behaves differently on either side of a point. For instance, a function might have a limit from the right but not from the left, or vice versa. These concepts are fundamental in calculus for defining continuity, derivatives, and integrals more rigorously.
🧮 Part A: Vocabulary
Match each term with its definition:
| Term |
Definition |
| 1. Limit from the Left |
A. The value $f(x)$ approaches as $x$ approaches $c$ only from values greater than $c$. |
| 2. Limit from the Right |
B. A function whose value remains constant over a specified interval. |
| 3. One-Sided Limit |
C. The value $f(x)$ approaches as $x$ approaches $c$ only from values less than $c$. |
| 4. Piecewise Function |
D. A function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. |
| 5. Constant Function |
E. The value a function approaches as the input approaches a specific value from either the left or the right side. |
✏️ Part B: Fill in the Blanks
Complete the following paragraph using the words provided: approaches, one-sided, value, direction, limit.
A ___________ limit considers the ___________ a function _______ as the input gets closer to a specific point from a single __________. This is different from a standard ________ which requires agreement from both sides.
🤔 Part C: Critical Thinking
Explain, in your own words, why understanding one-sided limits is important for analyzing the behavior of functions, especially those with discontinuities.