Determining Limits Using the Squeeze Theorem
Justify a trapping inequality and matching outer limits in a squeeze argument.
College Board Topic 1.8 · CED p. 40 (PDF p. 45) · CED effective Fall 2020; project course outline 2026–27.
Sometimes the middle function is awkward but its neighbors are easy. If g(x)≤f(x)≤h(x) for all inputs sufficiently close to c (except possibly c), and both outside functions approach the same finite L, then f(x) also approaches L. You must show both the trapping inequality and the matching outside limits.
Since −1≤sin(1/x)≤1, multiplying by x² gives −x²≤x²sin(1/x)≤x² for x≠0. Both bounds approach 0, so the middle limit is 0. The shrinking factor matters: sin(1/x) by itself has no limit at 0.
The geometric squeeze for 0<u<π/2 gives cos u≤sin u/u≤1. Both bounds approach 1 as u→0⁺; symmetry gives the same behavior from the left. This establishes sin u/u→1 in radians. The identity 1−cos u=2sin²(u/2) then gives (1−cos u)/u→0. These facts support trig derivatives later.
Let's work through it
- Start with |sin(1/x)|≤1, so |x²sin(1/x)|≤x² for x≠0. Reason: sine is bounded and x² is nonnegative.
- The lower bound −x² and upper bound x² both approach 0. Reason: power functions are continuous.
- Conclude the limit is 0. Reason: the squeeze theorem applies on a punctured neighborhood.
Try it from memory
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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 −|x|≤f(x)≤|x| near 0, what is lim(x→0)f(x)?
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 −|x|≤f(x)≤|x| near 0, what is lim(x→0)f(x)?
Worked answer and reason
0.
The two outside limits agree at 0.
Retrieval 2
If −1≤f(x)≤1 near 0, must f(x) approach 0?
Worked answer and reason
No.
The bounds approach different numbers, so the squeeze conclusion is unavailable.
Catch the tempting mistake
- A bounded function always has a limit.
- Bounded oscillation may have no limit; bounds must shrink to the same value.
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
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+ − * / ^ ! · 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 13 · No calculator · analytic
- Unit 1 problem 14 · No calculator · analytic
For the teacher
- Ask students to identify the neighborhood and the inequality before naming the theorem.
- Sketch narrowing bounds and then compare with constant-width bounds.
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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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.
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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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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.