The Squeeze Theorem is a powerful limit tool for functions that are hard to evaluate directly. It says that if a function is trapped between two other functions, and both outer functions approach the same value, then the trapped function must approach that value too. This matters because many oscillating or complicated expressions do not have obvious limits at first glance.
The theorem turns a difficult limit into a comparison problem that can often be solved with simple bounds.
A typical use is when one factor is bounded, such as -1 <= sin(1/x) <= 1, while another factor shrinks toward zero. Multiplying the bounds by a nonnegative expression like x^2 can force the whole product toward zero. The same idea also supports the famous result lim x->0 sin x / x = 1, where geometry shows that sin x / x is squeezed between two expressions with limit 1.
In graphs, the theorem looks like a sandwich, with the target function caught between an upper and lower curve that meet at the same point.
Key Facts
- If g(x) <= f(x) <= h(x) near a, and lim x->a g(x) = lim x->a h(x) = L, then lim x->a f(x) = L.
- The inequalities must hold on an interval around a, but they do not have to hold at x = a itself.
- Classic example: since -1 <= sin(1/x) <= 1, then -x^2 <= x^2 sin(1/x) <= x^2 for x near 0.
- Because lim x->0 -x^2 = 0 and lim x->0 x^2 = 0, lim x->0 x^2 sin(1/x) = 0.
- Important geometry result: lim x->0 sin x / x = 1, using radians.
- A bounded factor times a factor that approaches 0 also approaches 0: if |b(x)| <= M and lim x->a q(x) = 0, then lim x->a q(x)b(x) = 0.
Vocabulary
- Squeeze Theorem
- A limit theorem stating that a function trapped between two functions with the same limit must also have that limit.
- Bounding function
- A function used as an upper or lower comparison to control the possible values of another function.
- Limit
- The value a function approaches as the input approaches a specified number.
- Oscillation
- Repeated variation up and down, often making a function hard to analyze directly near a point.
- Radian
- A unit of angle measure required for standard calculus limits involving sine and cosine.
Common Mistakes to Avoid
- Forgetting to prove both inequalities. The Squeeze Theorem only applies after showing the target function really stays between the two bounding functions near the point.
- Using bounds that do not have the same limit. If the lower and upper functions approach different values, the trapped function is not forced to approach one specific number.
- Multiplying inequalities by a negative expression without reversing the signs. Inequality directions change when multiplied by a negative quantity, so the order of the bounds may become wrong.
- Applying lim x->0 sin x / x = 1 with degrees instead of radians. The standard calculus limit is true only when x is measured in radians.
Practice Questions
- 1 Use the Squeeze Theorem to find lim x->0 x^2 cos(5/x). Show the bounding inequalities.
- 2 Evaluate lim x->0 x sin(1/x) using the fact that -1 <= sin(1/x) <= 1.
- 3 Explain why the Squeeze Theorem can prove that x^2 sin(1/x) has a limit at 0 even though sin(1/x) itself does not have a limit at 0.