Key takeaways
- If g(x) ≤ f(x) ≤ h(x) near a, and g and h both approach L, then f approaches L too.
- cos(x) ≤ sin(x)/x ≤ 1 for small x ≠ 0, so lim(x→0) sin(x)/x = 1.
- The angle must be in radians. In degrees the limit is not 1.
- sin(0)/0 itself is undefined; only the limit equals 1.
In this guide
What is the squeeze theorem?
Imagine two friends walking on either side of you, holding your arms. If both of them walk through the same doorway, you have no choice: you go through that doorway too. That is the squeeze theorem, also called the sandwich theorem.
In maths: if g(x) ≤ f(x) ≤ h(x) near a point, and both g and h have the same limit L there, then f must also have the limit L.
Why does the squeeze theorem matter?
Some limits can't be found by substitution or algebra. The squeeze theorem finds them by comparison, using functions you already understand.
Its most famous result, sin(x)/x → 1, is the key step in proving that the derivative of sin(x) is cos(x). Without it, much of calculus with trig functions wouldn't work. You can watch the squeeze happen:
| h (radians) | cos(h) | sin(h)/h | Upper bound |
|---|---|---|---|
| 0.5 | 0.8776 | 0.9589 | 1 |
| 0.1 | 0.9950 | 0.9983 | 1 |
| 0.01 | 0.99995 | 0.99998 | 1 |
The two slices of bread close in on 1, and sin(h)/h is trapped in the middle.
The formula
| Symbol | What it means | Example |
|---|---|---|
| sin(x)/x | The function we want the limit of | 0.9983 at x = 0.1 |
| cos(x) | The lower bound; it approaches 1 | 0.9950 at x = 0.1 |
| 1 | The upper bound; always 1 | 1 |
| h | The small angle you enter, in radians | 0.1 |
The inequality comes from comparing three areas on a unit circle: a small triangle, a circle sector and a larger triangle. Both bounds head to 1, so sin(x)/x is squeezed to 1.
This calculator lets you enter a small value h and shows sin(h)/h next to the bounds cos(h) and 1. If you enter h = 0, it returns the exact limit, 1.
How to use the squeeze theorem step by step
- 1Find two bounds. Look for g(x) ≤ f(x) ≤ h(x) near the point.
- 2Find their limits. They are often easy, like polynomials or cos(x).
- 3Compare. If both limits equal the same number L, the limit of f is L too.
- 4Check numerically. Try h = 0.1, then 0.01, then 0.001, and watch the value approach L.
Now try some small angles yourself. Enter h below and see how close sin(h)/h is to 1.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Bounds with different limits. If g → 0 and h → 2, the squeeze tells you nothing.
- Forgetting where the inequality must hold. It only has to be true near the point, not everywhere.
- Thinking sin(0)/0 equals 1. The expression is undefined at 0; only the limit equals 1.
- Using bounds that cross. The lower function must stay below f and the upper one above it.
Tips and tricks
- Small-angle rule: since sin(x)/x → 1,
sin(x) ≈ xfor small angles in radians. - Related limit:
(1 − cos x)/x → 0as x → 0, which also follows from sin(x)/x → 1. - Next step: see how this limit gives the derivative in the derivative of sin calculator.
Where you'll use it in real life
- Calculus: proving the derivatives of sin and cos.
- Physics: the small-angle approximation sin θ ≈ θ for pendulums.
- Optics: simple lens formulas use the same approximation for small angles.
- Engineering: quick estimates when angles are small, such as a slight road slope.
Quick summary
The squeeze theorem traps a tricky function between two simpler ones with the same limit, forcing it to share that limit. With cos(x) ≤ sin(x)/x ≤ 1, it shows that sin(x)/x → 1 as x → 0, in radians. Use the calculator above to watch the squeeze for any small angle.
Worked example
You want to see the squeeze in action. Pick a small angle, h = 0.1 radians, and compare sin(h)/h with its bounds cos(h) and 1.
- 01Evaluating sin(h)/h at h = 0.1
- 02sin(0.1) = 0.0998
- 03sin(0.1) / 0.1 = 0.9983
- 04
- 05Squeeze Theorem bounds:
- 06cos(0.1) = 0.995 ≤ sin(h)/h ≤ 1
- 07As h → 0: both bounds → 1, so lim = 1
- 08At h=0.1: ratio = 0.9983 (close to 1)
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the limit of sin(x)/x as x approaches 0?
It is exactly 1, as long as x is measured in radians. For example, sin(0.1)/0.1 ≈ 0.9983.
Is the squeeze theorem the same as the sandwich theorem?
Yes. It is also called the pinching theorem. All three names describe trapping a function between two others.
How do I find the upper and lower functions?
Use facts you know, such as −1 ≤ sin(anything) ≤ 1, then multiply through. For x²·sin(1/x), this gives −x² ≤ x²·sin(1/x) ≤ x².
What happens if I enter h = 0 in the calculator?
It returns the exact limit, 1. For any other small h it shows sin(h)/h and the two bounds, so you can see how close you are.
