Selecting Procedures for Determining Limits
Select a limit procedure from the expression and direction of approach before calculating.
College Board Topic 1.7 · CED p. 39 (PDF p. 44) · CED effective Fall 2020; project course outline 2026–27.
Pause before calculating. The shape of an expression tells you which tool is likely to work. First try substitution where the functions are continuous. A finite nonzero denominator means the limit laws finish the job. An indeterminate 0/0 suggests factoring, a conjugate, or a trig reexpression. A boundary in a piecewise rule calls for left and right limits.
Unbounded behavior calls for sign analysis; bounded oscillation times a shrinking factor often calls for a squeeze. Graphs and tables help identify the behavior, but state whether they give an estimate or a proof. When two methods are possible, choose the one with the clearest justified steps.
Compare three limits at x=0: sin(2x)/x needs a trig reexpression; (√(1+x)−1)/x suggests a conjugate; 1/x needs one-sided sign analysis. All have a denominator approaching 0, so the denominator alone cannot select the procedure.
Let's work through it
- Consider lim(x→2)(x²−4)/(x−2). Substitution gives 0/0. Reason: the quotient limit theorem does not settle a zero denominator.
- Choose factoring rather than a decimal table. Reason: the numerator is a difference of squares.
- Cancel for x≠2, then take the limit of x+2 to get 4. Reason: an equivalent nearby expression is continuous.
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
Which first method fits lim(x→1)(x²+3)/(x+2)?
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
Which first method fits lim(x→1)(x²+3)/(x+2)?
Worked answer and reason
Direct substitution with the quotient limit rule.
The denominator approaches 3, so no algebraic repair is needed.
Retrieval 2
Which first method fits lim(x→0)x²cos(1/x)?
Worked answer and reason
The squeeze theorem.
The oscillating factor is bounded and x² shrinks to 0.
Catch the tempting mistake
- Every zero denominator means factor the numerator.
- A pole, an oscillation and a removable hole require different arguments.
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 12 · No calculator · analytic
For the teacher
- Give a card sort where students name the method without completing the calculation.
- Use this selection checkpoint again after students learn more limit methods.
Glossary and deeper reading · 2 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 laws
Limit laws allow limits of sums, products, and suitable quotients to be assembled from the limits of their parts. The component limits must exist as required, and a quotient law needs a nonzero denominator limit. For example, if f approaches 2 and g approaches 3, then f+g approaches 5 and fg approaches 6. Direct substitution into a quotient with denominator limit 0 is not justified by the quotient limit law.
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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.