Squeeze Theorem Applier

Math Calculus • Limits and Continuity

Written by STEM Calculators Team Published December 21, 2025 Updated February 24, 2026

6. Squeeze Theorem Applier

Apply the Squeeze Theorem by entering bounds \(g(x)\le f(x)\le h(x)\) near the approach point. If \(\lim g=\lim h=L\), then \(\lim f=L\).

Inputs

Target function \(f(x)\)

Supported: + − * / ^, parentheses, x, pi, e, sin cos tan, ln log (base 10), sqrt abs exp. Implicit multiplication allowed: 2x, (x+1)(x-1), 2sin(x).

Limit mode

Squeeze is typically used for finite \(a\), but infinity modes are supported numerically.

Approach point \(a\)

Accepts constants like pi, e, and expressions like 2*pi. (Ignored for \(\pm\infty\) modes.)

Lower bound \(g(x)\)

Enter a function intended to satisfy \(g(x)\le f(x)\) near the approach point.

Upper bound \(h(x)\)

Enter a function intended to satisfy \(f(x)\le h(x)\) near the approach point.

Options Verify inequality numerically Show sandwich shading (between \(g\) and \(h\)) Show grid

Approach depth

Uses \(a\pm10^{-k}\), \(k=1..N\) (finite \(a\)).

Graph window (half-width)

For finite \(a\): \(x\in[a-w,a+w]\).

Inequality tolerance

Allows small numeric error when checking \(g\le f\le h\).

Tools

Suggests common squeezes for trig patterns (e.g. \(|\sin|\le1\)).

Presets

Click to load an example (rendered with MathJax).

Ready

Graph sandwich

Drag to pan • wheel/pinch to zoom • dashed line is \(x=a\) (finite \(a\)). Curves: \(g\) (lower), \(f\) (target), \(h\) (upper).

x: 0, y: 0, zoom: 1

Result

Enter \(f(x)\), bounds \(g(x)\), \(h(x)\), then click Apply Squeeze.

Rate this calculator

0.0 /5 (0 ratings)

Be the first to rate.

Your rating

Name (optional) Review (optional)

You can update your rating any time.

6. Squeeze Theorem Applier — Theory

The Squeeze Theorem lets you evaluate a limit of a difficult function by trapping it between two easier functions. The idea is purely order-based: if \(f(x)\) is always between \(g(x)\) and \(h(x)\) near the approach point, and the bounds meet at the same limit, then \(f(x)\) must meet that limit too.

1) Statement of the Squeeze Theorem

\[ \text{If } g(x)\le f(x)\le h(x) \text{ for all } x \text{ in a punctured neighborhood of } a, \] \[ \text{and } \lim_{x\to a} g(x)=\lim_{x\to a} h(x)=L, \text{ then } \lim_{x\to a} f(x)=L. \]

“Punctured neighborhood” means: the inequality must hold for \(x\) close to \(a\) but not necessarily at \(x=a\) itself.

2) How you choose bounds

Bounds often come from simple universal inequalities, especially for trig:

\(-1 \le \sin(\theta) \le 1\) and \(-1 \le \cos(\theta) \le 1\)
\(|\sin(\theta)| \le 1\) and \(|\cos(\theta)| \le 1\)
If \(|u(x)|\le 1\), then \(-|A(x)|\le A(x)\,u(x)\le |A(x)|\)

3) Worked example: \(\lim_{x\to 0} x\sin(1/x)\)

Since \(-1\le \sin(1/x)\le 1\), multiplying by \(|x|\) gives:

\[ -|x| \le x\sin(1/x) \le |x|. \]

Now compute the bounds:

\[ \lim_{x\to 0}(-|x|)=0, \qquad \lim_{x\to 0}|x|=0. \]

Both bounds approach \(0\), so the squeezed function must also approach \(0\):

\[ \boxed{\lim_{x\to 0} x\sin(1/x)=0.} \]

4) Graph intuition

If the graph of \(f(x)\) stays inside the “sandwich” region between \(g(x)\) and \(h(x)\), and the sandwich closes to a single height \(L\) as \(x\to a\), then \(f(x)\) is forced to go to \(L\) as well.

5) Common squeeze patterns

\(|\sin(1/x)|\le 1 \Rightarrow -|x|^n \le x^n\sin(1/x)\le |x|^n\) (limit \(0\) if \(n>0\))
\(|\cos(1/x)|\le 1 \Rightarrow -|x|^n \le x^n\cos(1/x)\le |x|^n\)
Classic: for \(x\ne0\) near \(0\), \(\cos x \le \frac{\sin x}{x} \le 1\), hence \(\lim_{x\to0}\frac{\sin x}{x}=1\)

6) Notes and limitations

The squeeze theorem requires the bounds’ limits to be the same finite value \(L\).
If your inequality fails (even on one side), you cannot conclude.
This tool verifies the inequality numerically (sample points). A passing check is strong evidence, but it is not a formal proof by itself.

Frequently Asked Questions

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

Find functions g(x) and h(x) such that g(x) <= f(x) <= h(x) near the approach point and the limits of g and h match. If lim g = lim h = L, then lim f = L.

What happens if the inequality g(x) <= f(x) <= h(x) is not satisfied?

If the inequality fails near the approach point, the squeeze theorem cannot be used to conclude the limit. You may need different bounds or a different limit method.

How does the calculator verify the squeeze conditions?

It samples x-values near the approach point using a +/- 10^(-k) and checks whether g(x) <= f(x) <= h(x) within a chosen numeric tolerance. It also computes the limits of g and h in the selected limit mode.

Can the squeeze theorem be used for limits as x approaches infinity?

Yes, if you can bound f(x) between two functions whose limits at infinity are the same. This calculator supports infinity modes numerically to help confirm the behavior.