Connecting Multiple Representations of Limits
Translate one limit among a formula, graph, table, and verbal statement with consistent units.
College Board Topic 1.9 · CED p. 41 (PDF p. 46) · CED effective Fall 2020; project course outline 2026–27.
A useful limit argument can move between a story, a table, a graph and a formula. Keep the same input target, output target, units, and direction of approach while translating. The graph and table should tell the same nearby-output story as the analytic expression.
Take a device whose signal is R(t)=(t²−9)/(t−3) volts for t≠3 seconds. Algebra gives R(t)=t+3 nearby. A table at 2.99 and 3.01 gives 5.99 and 6.01; the graph is a line with a hole at (3,6); the verbal conclusion is “the signal approaches 6 volts as time approaches 3 seconds.” The device may report no reading at exactly 3 seconds.
If representations disagree, investigate rather than choosing your favorite. A rounded table may hide a jump, a graph window may conceal oscillation, and a formula may be entered incorrectly. Explain what each representation establishes and what still requires a neighborhood argument.
Let's work through it
- Simplify R(t) to t+3 for t≠3. Reason: factoring yields an equivalent expression nearby.
- Predict a line approaching height 6 at t=3 and a table tending toward 6 from both sides. Reason: all representations describe the same rule.
- Write lim(t→3)R(t)=6 V, without assigning R(3). Reason: approaching output and actual target output are distinct.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
A table tends to 5 but the filled graph point is at height 8. Is there a contradiction?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
A table tends to 5 but the filled graph point is at height 8. Is there a contradiction?
Worked answer and reason
Not necessarily; the limit may be 5 while the function value is 8.
Nearby behavior differs from the value at the target.
Retrieval 2
What unit belongs to a limit of temperature T(t)?
Worked answer and reason
Temperature units, such as degrees Celsius.
A function limit is an output value; a rate would have temperature per time units.
Catch the tempting mistake
- Every representation gives an exact proof by itself.
- A finite table or image usually estimates; analytic reasoning can justify the complete local behavior.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 1 problem 15 · No calculator · multiple
For the teacher
- Have groups produce four representations of one removable hole.
- Require a sentence about evidence strength when students use a calculator table.
Glossary and deeper reading · 1 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.