What is the squeeze theorem?

The squeeze theorem (also called the pinching theorem or sandwich theorem) states that if f(x) ≤ g(x) ≤ h(x) near a point a, and both f(x) and h(x) converge to the same limit L as x approaches a, then g(x) must also converge to L. This is useful for evaluating limits of functions that oscillate or are difficult to analyze directly.

What is an example of the squeeze theorem?

A classic example is lim(x→0) x·sin(1/x) = 0. We observe that -|x| ≤ x·sin(1/x) ≤ |x| for all x ≠ 0. Since lim(x→0) -|x| = 0 and lim(x→0) |x| = 0, the squeeze theorem implies lim(x→0) x·sin(1/x) = 0. This calculator can verify this directly.

How do I use the squeeze theorem to evaluate a limit?

To use the squeeze theorem to evaluate a limit: (1) Identify two bounding functions f(x) and h(x) that 'squeeze' your target function g(x). (2) Verify algebraically (or numerically with this tool) that f(x) ≤ g(x) ≤ h(x) in a neighborhood of the limit point. (3) Show that lim f(x) = lim h(x) = L. (4) Conclude that lim g(x) = L. This calculator automates the numerical verification step.

What is the sandwich theorem?

The sandwich theorem is another name for the squeeze theorem. It uses the metaphor of a function being 'sandwiched' between two bounding functions. In some textbooks, particularly those using European terminology, it may also be called the compression theorem or squeezing theorem. All names refer to the same principle.

Can I use the squeeze theorem for sequences?

Yes! The squeeze theorem applies equally to sequences. If a_n ≤ b_n ≤ c_n for all n, and both lim(n→∞) a_n and lim(n→∞) c_n equal L, then lim(n→∞) b_n = L. Select 'Sequences' in this calculator to verify squeeze theorem conditions on discrete sequences.

What does it mean when the calculator says 'Inconclusive'?

The result is 'Inconclusive' if the numerical sampling indicates that either (a) the bounds do not appear to converge to the same limit within the tested neighborhood, or (b) you did not provide estimated limits for the bounding functions. Try expanding the sample count, narrowing δ, or supplying explicit limit values L_f and L_h.

What is a proof of the squeeze theorem?

Proof outline: Let ε > 0. If lim f(x) = L and lim h(x) = L as x→a, then there exist δ₁, δ₂ > 0 such that |f(x) − L| < ε and |h(x) − L| < ε when 0 < |x − a| < δ. Set δ = min(δ₁, δ₂). Since f(x) ≤ g(x) ≤ h(x), we have L − ε < f(x) ≤ g(x) ≤ h(x) < L + ε, so |g(x) − L| < ε. Thus lim g(x) = L. This calculator provides numerical evidence supporting this reasoning.

What is the sinx/x squeeze theorem example?

One famous limit is lim(x→0) sin(x)/x = 1. Using the pinching theorem: for small x > 0, we have cos(x) < sin(x)/x < 1/cos(x). Alternatively, observe that |sin(x)| ≤ |x| ≤ sec(x)·sin(x) near 0, leading to lim sin(x)/x = 1. This classic limit is difficult without the sandwich theorem and is included as a preloaded example.

What are common squeeze theorem mistakes?

Common pitfalls include: (1) Forgetting to verify that both bounds actually converge to the same limit. (2) Bounds that only hold at a single point, not in a neighborhood. (3) Assuming the middle function converges without checking the bounds. (4) Using asymmetric bounds when the limit point lies at the boundary. This calculator's numeric checks help catch these errors.

Can I download squeeze theorem practice problems?

Yes! This calculator includes preloaded practice problems covering functions, sequences, and oscillatory examples. Click 'Load Example' to explore curated problems. Export results to CSV for your records or to build a personal worksheet. The 'Show Steps' feature provides detailed solutions.

How does this squeeze theorem solver work?

The calculator: (1) Parses your input functions or sequences safely. (2) Numerically samples f, g, h (or a_n, b_n, c_n) at points near the limit (or over the index range). (3) Verifies whether f ≤ g ≤ h at each sampled point. (4) Estimates the limit by symmetric sampling and robust averaging. (5) Concludes whether squeeze theorem applies. Results are displayed with diagnostic points and confidence indicators.

Can I use this squeeze theorem calculator for homework and exams?

Yes, this calculator is a learning tool. Use it to verify your work, understand why a limit is squeezed to a particular value, and explore examples of the sandwich theorem in action. However, for exams, focus on understanding the reasoning—this tool helps you check your algebra and gain intuition, not replace mathematical rigor.

Squeeze (Sandwich) Theorem Calculator & Solver

Verify the pinching theorem for functions, limits, and sequences with step-by-step analysis

Squeeze Theorem (Functions & Limits):

If f(x) ≤ g(x) ≤ h(x) near a point and lim(x→a) f(x) = lim(x→a) h(x) = L, then lim(x→a) g(x) = L.

Squeeze Theorem (Sequences):

If aₙ ≤ bₙ ≤ cₙ for all n, and lim(n→∞) aₙ = lim(n→∞) cₙ = L, then lim(n→∞) bₙ = L.

Evaluation Type

Input & Configuration

Lower Bound f(x)

Expression in x (e.g., -1, -abs(x), cos(x))

Middle Function g(x)

Your target function

Upper Bound h(x)

Expression in x

Limit Point a

Where x → a

Interval (δ)

Sample range around a

Sample Count

Points to check (higher = more precise)

Limit L_f (optional)

Estimated lim f(x) as x → a

Limit L_h (optional)

Estimated lim h(x) as x → a

Quick Examples:

Lower Sequence aₙ

Expression in n (e.g., 1/n, (2*n)/(n+1))

Middle Sequence bₙ

Your target sequence

Upper Sequence cₙ

Expression in n

Starting Index n₀

Begin sampling from n =

Index Window

Sample up to n =

Sample Count

Number of terms to check

Limit L_lower (optional)

Estimated lim aₙ as n → ∞

Limit L_upper (optional)

Estimated lim cₙ as n → ∞

Quick Examples:

Run Verification

Results

Results will appear here after verification.

Squeeze (Sandwich) Theorem Calculator & Solver

Evaluation Type

Input & Configuration

Run Verification

Results

How It Works (Steps)

Verification Steps

Practice & Examples