1.9 Connecting Multiple Representations of Limits
- Syllabus
- 2020
- Topic
- 1.9
- Level
- —
Learning objectives
Translating the Same Limit Across Representations
A correct translation keeps the same target input, direction of approach, approached output, and level of certainty. Only the representation changes; the mathematical claim does not.
\lim_{x\to2}f(x)=5
| Representation | Meaning of the same claim |
|---|---|
| Analytical | limx→2f(x)=5 |
| Verbal | As inputs approach 2, the outputs approach 5 |
| Numerical | For inputs increasingly close to 2 from below and above, table values approach 5 |
| Graphical | The graph's height approaches 5 as x approaches 2 from both sides |
For f(x)=x+3, a table gives f(1.99)=4.99 and f(2.01)=5.01. The values approach 5 from both sides of 2, so the analytical re-expression is limx→2(x+3)=5; verbally, outputs approach 5 as inputs approach 2.
Before accepting a translation, check: the same target x-value; the same left, right, or two-sided direction; the same approached y-value; and whether the source supports an exact value or only an estimate.
A filled point at (2,5) states f(2)=5, not by itself the limit. Likewise, evidence from only the left cannot justify a two-sided limit. Translate nearby behavior and direction, not merely the function's value at the target.