Estimating Limit Values from Graphs
Read nearby graph behavior, holes, jumps, oscillation, and unbounded output without mistaking a picture for proof.
College Board Topic 1.3 · CED p. 35 (PDF p. 40) · CED effective Fall 2020; project course outline 2026–27.
Trace a graph toward the target from the left and right. Follow the curve, not a lone filled dot. Record the two one-sided heights separately before claiming a two-sided limit.
A graph can suggest a limit but a plotting window can hide a narrow spike, a hole or fast oscillation. Zooming helps; an analytic rule settles behavior when a finite picture cannot. At x=0, sin(1/x) keeps oscillating between −1 and 1; no single nearby height takes over. For 1/x² the heights grow without bound; “+∞” describes that growth, rather than a finite real limit. For 1/x, the two sides have opposite signs of unbounded growth.
Let's work through it
- In the pictured graph, both nearby traces approach height 3. Reason: the open point marks the height approached from either side.
- The filled point gives f(2)=5. Reason: filled points indicate included function values.
- Conclude lim(x→2)f(x)=3. Reason: both one-sided limits agree, independent of the filled point.
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 left trace approaches 1 and a right trace approaches 4. What is the two-sided limit?
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 left trace approaches 1 and a right trace approaches 4. What is the two-sided limit?
Worked answer and reason
It does not exist.
Unequal one-sided limits produce a jump.
Retrieval 2
A screen draws sin(1/x) as a nearly solid band near 0. Does that prove a limit of 0?
Worked answer and reason
No; the function continues to oscillate.
A plot can miss infinitely rapid behavior; the formula supplies the counterexample.
Catch the tempting mistake
- The highest visible point near a target is the limit.
- A limit concerns every sufficiently nearby input, rather than one visible high point.
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 4 · No calculator · graphical
- Unit 1 problem 5 · No calculator · multiple
For the teacher
- Read graphs at two scales and ask what each scale could hide.
- Keep a complete text description beside every graph for students using screen readers.
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.