Limit Using Squeeze Theorem Calculator

Calculator

Enter a lower bound, the middle function, and an upper bound. The tool samples both sides of the target point, checks ordering, estimates common behavior, and plots all three curves.

Lower bound L(x)

Example: -abs(x), x^2 - x^4, -x^2

Middle function M(x)

Example: x*sin(1/x), x^2*cos(1/x), (sin(x))/x

Upper bound U(x)

Example: abs(x), x^2 + x^4, x^2

Target point a

Point where x approaches a.

Expected common limit

Usually the shared bound limit candidate.

Sampling window

Range from a-window to a+window.

Sample points

Higher values create smoother checks and graph.

Tolerance

Allowed numeric difference for near-limit testing.

Excluded radius around a

Avoids evaluating exactly at singular points.

Plotly graph

The chart compares lower, middle, and upper functions around the target point. A vertical guide marks the point being approached.

Computed sample table

#	x	L(x)	M(x)	U(x)	Inside bounds?	\|L(x)-Limit\|	\|U(x)-Limit\|
Submit the form to generate values.

Example data table

This example uses the classic squeeze theorem setup for x sin(1/x) as x → 0.

x	-\|x\|	x sin(1/x)	\|x\|	Observation
-0.2000	-0.2000	-0.1918	0.2000	Middle stays between both bounds.
-0.1000	-0.1000	0.0544	0.1000	Oscillates, but magnitude remains bounded.
0.1000	-0.1000	-0.0544	0.1000	Closer to zero as x nears zero.
0.0500	-0.0500	0.0456	0.0500	Bounds and middle all trend toward zero.

Formula used

The squeeze theorem states that if three functions satisfy L(x) ≤ M(x) ≤ U(x) for all x near a, except possibly at a itself, and if both outer functions approach the same limit A, then the middle function also approaches A.

If L(x) ≤ M(x) ≤ U(x) for x near a

and lim x→a L(x) = lim x→a U(x) = A

then lim x→a M(x) = A

This calculator uses numeric sampling near the target point, checks the inequality across valid samples, and compares the outer functions with the expected common limit. It is a practical verification tool, not a symbolic proof engine.

How to use this calculator

Enter a lower bound that stays less than or equal to the middle function near the target point.
Enter the middle expression whose limit you want to study.
Enter an upper bound that stays greater than or equal to the middle function nearby.
Set the target point, expected shared limit, sampling window, and tolerance.
Click Calculate limit to produce the graph, table, and squeeze assessment.
Use CSV or PDF download buttons to save the computed sample values and summary.

FAQs

1) What does this calculator actually verify?

It numerically checks whether the middle function remains between a lower and upper bound near a chosen point, and whether both bounds appear to approach the same limit.

2) Can this replace a formal proof?

No. It provides strong numeric evidence and visualization, but a rigorous proof still requires valid analytical reasoning about inequalities and limits.

3) Why exclude a small radius around the target point?

Some squeeze theorem problems involve expressions undefined at the target point, like sin(1/x) at x = 0. The excluded radius avoids direct evaluation there.

4) What expressions are supported?

Common math expressions supported by math.js work well, including abs, sin, cos, tan, sqrt, log, exp, powers, and combinations using the variable x.

5) What if the calculator says the bound check fails?

That means at least one tested sample did not satisfy L(x) ≤ M(x) ≤ U(x). Your chosen bounds may be incorrect, or the window may be too wide.

6) How should I choose the expected limit?

Use the value both outer bounds are supposed to approach. In many classic problems, the lower and upper bounds are symmetric and squeeze the middle function to zero.

7) Why does the graph matter?

The graph helps you see whether the middle curve remains trapped between the outer curves and whether all three move toward the same value as x approaches the target.

8) Can I use this for one-sided thinking?

Yes, approximately. Use a smaller window and inspect values from one side in the sample table, though the tool mainly performs two-sided numeric checking.