Determining Limits Using Algebraic Manipulation
Use factors, conjugates, and trigonometric identities to preserve nearby values while resolving a limit.
College Board Topic 1.6 · CED p. 38 (PDF p. 43) · CED effective Fall 2020; project course outline 2026–27.
When substitution gives 0/0, read that as “the first method did not settle this,” not as an answer. Look for a formula equal to the original at nearby inputs. Factoring, conjugates, and trigonometric identities often expose a removable factor.
For (x²−9)/(x−3), factoring gives (x−3)(x+3)/(x−3)=x+3 for x≠3. That restriction is exactly right for a limit at 3. For (√(x+1)−1)/x, multiply numerator and denominator by √(x+1)+1. The new quotient is 1/(√(x+1)+1) for x≠0, so its limit at 0 is 1/2.
For trig limits use radians. The standard limit sin u/u→1 as u→0 lets us rewrite sin(5x)/x=5·sin(5x)/(5x), which approaches 5. Cancellation is multiplication and division of factors, not deletion of terms inside a sum. L’Hôpital’s rule is introduced later after derivatives; it is not needed for these examples.
Let's work through it
- Factor x²−9=(x−3)(x+3). Reason: a difference of squares exposes the common factor.
- Cancel x−3 for x≠3. Reason: the simplified expression agrees on the inputs used in the limit.
- Evaluate the limit of x+3 at 3 as 6. Reason: this remaining function 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
Evaluate lim(x→4)(x²−16)/(x−4).
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
Evaluate lim(x→4)(x²−16)/(x−4).
Worked answer and reason
8.
Factor and cancel a common factor away from x=4.
Retrieval 2
Evaluate lim(x→0)sin(3x)/x in radians.
Worked answer and reason
3.
Reexpress it as 3 times sin(3x)/(3x).
Catch the tempting mistake
- 0/0 means the limit equals 0.
- 0/0 is indeterminate: different functions with this substitution can have different limits or no limit.
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 9 · No calculator · analytic
- Unit 1 problem 10 · No calculator · analytic
- Unit 1 problem 11 · No calculator · analytic
- Unit 1 problem 12 · No calculator · analytic
For the teacher
- Ask students to annotate the excluded target after cancellation.
- Contrast cancellation of factors with invalid cancellation in (x+2)/(x+3).
Glossary and deeper reading · 4 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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Trigonometric identity
A trigonometric identity is an equality that holds for every input where both sides are defined. Identities let you rewrite expressions without changing their values on that shared domain. For example, sin²θ+cos²θ=1 follows from the unit circle. Any division used while rewriting an identity can introduce domain restrictions.
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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.
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Radian
A radian measures an angle by dividing the intercepted arc length by the circle’s radius. One full turn is 2π radians, so π radians equals 180 degrees. For example, an arc with length equal to its radius subtends an angle of one radian. The familiar derivative formulas for sine and cosine use radian inputs.
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L’Hospital’s rule
L’Hospital’s rule can evaluate certain quotient limits with the indeterminate forms 0/0 or infinity divided by infinity. Its hypotheses require appropriate differentiability near the target, a nonzero denominator derivative there, and a suitable limit of the derivative quotient. For example, lim(x→0) sin x/x equals lim(x→0) cos x/1=1. Differentiating numerator and denominator separately here is a limit theorem, not the quotient rule for derivatives.