Calculator Inputs
Formula Used
The squeeze theorem states that if g(x) ≤ f(x) ≤ h(x) near a,
except possibly at a, and lim g(x) = lim h(x) = L,
then lim f(x) = L.
This calculator checks sampled inequalities and estimates bound convergence with
gap = |h(x) - g(x)|. When the near-bound gap is within tolerance,
the common limit estimate is (g_near + h_near) / 2.
How to Use This Calculator
- Enter the lower bound, target function, and upper bound using x as the variable.
- Set the value that x approaches and choose the approach direction.
- Adjust the outer and inner radii to control the sampling zone.
- Choose a tolerance that matches the engineering precision you need.
- Press calculate, then review the conclusion, chart, table, CSV, and PDF.
Example Data Table
| Engineering case | g(x) | f(x) | h(x) | a | Expected limit |
|---|---|---|---|---|---|
| Oscillating vibration envelope | -abs(x) | x*sin(1/x) | abs(x) | 0 | 0 |
| Signal ripple damping | -x^2 | x^2*cos(1/x) | x^2 | 0 | 0 |
| Bounded thermal disturbance | 5-abs(x-2) | 5+(x-2)*sin(1/(x-2)) | 5+abs(x-2) | 2 | 5 |
Engineering Article
Why the Squeeze Theorem Matters
Engineering models often include oscillating terms, rapid switching, and small disturbances. A direct limit may look unclear because the central expression changes sign many times. The squeeze theorem gives a dependable path. It surrounds the difficult function with two simpler bounds. When both bounds approach the same value, the trapped model must approach that value too.
How the Method Works
The calculator uses three expressions. The lower bound is g(x). The target function is f(x). The upper bound is h(x). The required condition is g(x) ≤ f(x) ≤ h(x) near the approach value. The bounds do not need to equal the target. They only need to trap it while moving toward one shared limit.
Numerical Sampling Near the Point
This tool samples points around the approach value. It avoids the exact point when the formula may be undefined there. The outer radius controls the wider testing region. The inner radius checks behavior very close to the limit. More samples give smoother evidence. Smaller tolerances create stricter conclusions. The gap between upper and lower bounds is the most important numerical signal.
Engineering Interpretation
The theorem is useful for vibration envelopes, ripple decay, signal noise, stability margins, and thermal perturbation models. It helps show that a bounded oscillation becomes harmless when its envelope shrinks. A result near zero can mean a transient disappears. A result near another value can show steady behavior around an operating point.
Good Practice
Use bounds that are physically meaningful. Check units before trusting the result. Try both one-sided and two-sided settings when the model changes across a threshold. If the pass rate is low, review the inequality or widen the tolerance only when justified. Numerical output supports judgment, but final engineering proof should include symbolic reasoning.
Exporting Results
The CSV file is helpful for spreadsheets and reports. The PDF file is useful for design notes. The plot gives a quick visual check. Use all three outputs together when documenting assumptions, tolerance choices, sample density, and the final limit estimate for review teams or future calculations. This makes the calculation easier to audit during engineering approval workflows.
FAQs
1. What does this calculator estimate?
It estimates whether a function limit can be supported by the squeeze theorem. It checks sampled bounds, compares gaps, and gives an approximate common limit.
2. Can it prove a limit completely?
No. It gives strong numerical support. A complete proof still needs valid symbolic inequalities over an interval near the approach value.
3. Which variable should I use?
Use x as the variable. The calculator reads expressions like sin(x), x^2, abs(x), exp(x), log(x), and x*sin(1/x).
4. What does tolerance mean?
Tolerance is the allowed numerical error. Smaller tolerance makes the test stricter. Use a value that matches your engineering precision.
5. Why are inner and outer radii needed?
They define the sampling zone around the approach value. Smaller inner radius checks behavior closer to the limit point.
6. What happens if samples are invalid?
Invalid points are skipped. This may happen with division by zero, logarithms of non-positive values, or square roots of negatives.
7. Can I test one-sided limits?
Yes. Choose left-hand or right-hand direction. This helps when an engineering model behaves differently on each side.
8. Why is the graph useful?
The graph shows whether the target curve remains trapped between bounds. It also reveals weak bounds, asymmetry, or poor radius choices.