Determining Limits Using Algebraic Properties of Limits
Apply limit laws and composition only when their denominator and continuity conditions hold.
College Board Topic 1.5 · CED p. 37 (PDF p. 42) · CED effective Fall 2020; project course outline 2026–27.
When component limits are finite, limits pass through addition, subtraction, multiplication and constant multiples. For a quotient, the limiting denominator must be nonzero. That condition earns its place: dividing by a value approaching 0 can produce wildly different outcomes.
If g(x) approaches L and f is continuous at L, then f(g(x)) approaches f(L). Continuity of the outside function is the permission slip. Direct substitution is therefore reliable for polynomial functions and for familiar continuous functions at points in their domains. A piecewise boundary still needs one-sided checks.
For example, as x→1, x²→1 and 3x+2→5, so (x²+2)/(3x+2)→3/5. By contrast, direct substitution in (x²−1)/(x−1) gives 0/0: the quotient theorem has not applied, and a new method is needed.
Let's work through it
- For lim(x→1)(x²+2)/(3x+2), find numerator limit 3 and denominator limit 5. Reason: polynomials are continuous.
- Check 5≠0. Reason: the quotient limit rule requires a nonzero limiting denominator.
- Divide to obtain 3/5. Reason: all conditions of the quotient rule for limits hold.
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
If f→2 and g→−3, what does 4f−g approach?
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
If f→2 and g→−3, what does 4f−g approach?
Worked answer and reason
11.
Use the constant multiple and difference rules.
Retrieval 2
Can the quotient limit rule divide by a limiting denominator of 0?
Worked answer and reason
No.
That hypothesis fails; cancellation or one-sided analysis may be needed.
Catch the tempting mistake
- Substitution is a universal rule for limits.
- It follows from continuity or valid limit laws; it can fail at a hole or piecewise boundary.
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 7 · No calculator · analytic
- Unit 1 problem 8 · No calculator · analytic
For the teacher
- Require the nonzero-denominator check in written solutions.
- Include a composition whose outside function is discontinuous so students see why its hypothesis matters.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
-
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.
-
Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
-
Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.