In pre-calculus, graphical limits help us understand a function's behavior as it approaches a specific x-value. Instead of algebraic manipulation, we analyze the graph to determine what y-value the function approaches from both the left and right sides. If both sides approach the same y-value, the limit exists at that point. If the function approaches different y-values or doesn't approach any value at all, the limit does not exist. Understanding graphical limits is crucial for grasping continuity and derivatives later on!
Match the term with its correct definition:
| Term | Definition |
|---|---|
| 1. Limit | A. A value that a function approaches as the input approaches some value. |
| 2. Asymptote | B. A line that a curve approaches, without ever actually touching it. |
| 3. Discontinuity | C. A point at which a function is not continuous. |
| 4. One-Sided Limit | D. The value a function approaches from either the left or the right side. |
| 5. Indeterminate Form | E. An expression whose limit cannot be evaluated directly by substitution. |
A _____ exists at a point if the function approaches the same value from both the left and the _____. A _____ is a line that the function gets closer and closer to, but never touches. If a function has a jump, hole, or vertical asymptote at a certain point, it has a _____ there. When evaluating limits graphically, we look at the _____ values as we get closer to a specific x-value.
Explain in your own words how you can determine if a limit exists at a point on a graph. What conditions must be met?
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