Squeeze Theorem

Core Definition

The Squeeze Theorem (also known as the Sandwich Theorem or Pinching Theorem) states that if:

$g(x) \le f(x) \le h(x)$ for all $x$ in an open interval containing $c$ (except possibly at $c$ itself), and
$\lim_{x \to c} g(x) = L$ and $\lim_{x \to c} h(x) = L$ ,

then the trapped middle function must also converge to the exact same limit:

\lim_{x \to c} f(x) = L

Concept Explanation

The Sandwiching Mechanism

When a function $f(x)$ exhibits high-frequency oscillations or complex analytical forms near a boundary point $c$ , direct substitution and standard algebraic simplifications often fail. If two simpler bounding functions $g(x)$ (lower bound) and $h(x)$ (upper bound) can be constructed such that:

$f(x)$ is strictly sandwiched between $g(x)$ and $h(x)$ , and
$g(x)$ and $h(x)$ squeeze together to the same target value $L$ as $x \to c$ ,

then the geometry forces $f(x)$ into that same target $L$ without requiring direct algebraic resolution of $f(x)$ .

   y ^                h(x) (Upper Bound)
     |          \      /
     |           \    /  <-- Pinched together at (c, L)
   L |------------*
     |           / \     f(x) (Trapped oscillating function)
     |          /   \
     |               g(x) (Lower Bound)
     +---------------------> x
                   c

Bounding Oscillating Functions

The most common application of the Squeeze Theorem in calculus involves trigonometric functions whose arguments shoot to infinity near zero, such as $\sin(1/x)$ or $\cos(1/x)$ .

Because the range of sine and cosine is strictly bounded by $[-1, 1]$ :

-1 \le \sin\left(\frac{1}{x}\right) \le 1 \quad \text{and} \quad -1 \le \cos\left(\frac{1}{x}\right) \le 1 \quad (\text{for all } x \neq 0)

Multiplying this fundamental inequality by a damping envelope factor $x^{2n}$ (or $|x|$ ) creates the sandwich bounds (see Solving Inequalities).

Engineering Applications

Transient Structural Vibration Damping (Civil & Mechanical Engineering): When an earthquake or sudden mechanical shock impacts a structure, a tuned mass damper produces transient lateral sway modeled by $x(t) = t^2 \sin(\omega / t)$ . Applying the Squeeze Theorem proves that the structural displacement converges smoothly to zero as $t \to 0^+$ without infinite velocity shocks.
Optical Laser Telemetry & Diffraction (Entertainment & Computer Engineering): Laser scanning telemetry models beam envelope interference across diffraction slits. Sandwiched inequality envelopes prove beam energy thresholds remain bounded at micro-aperture centers.

Example Problems

Example 1: Damped High-Frequency Oscillation

Evaluate the limit:

\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right)

Solution

Step 1: Establish the trigonometric bound: For all $x \neq 0$ , the sine function is bounded:

-1 \le \sin\left(\frac{1}{x}\right) \le 1

Step 2: Multiply by the positive damping term $x^2$ : Since $x^2 > 0$ for all $x \neq 0$ , multiplying preserves the inequality directions:

-x^2 \le x^2 \sin\left(\frac{1}{x}\right) \le x^2

Step 3: Evaluate the limits of the bounding functions:

\lim_{x \to 0} (-x^2) = 0 \quad \text{and} \quad \lim_{x \to 0} (x^2) = 0

Step 4: Apply the Squeeze Theorem: Since the lower and upper bounds both approach $0$ , the trapped middle function must also approach $0$ :

\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0

Example 2: Inequality Bounds on an Unknown Function

Suppose a sensor signal $h(t)$ satisfies $2t^3 + 8 \le h(t) \le t^4 + 8$ for all $t$ near $2$ . Find $\lim_{t \to 2} \frac{h(t) + 5}{3}$ .

Solution

Step 1: Evaluate the lower and upper bound limits as $t \to 2$ :

Lower bound: $\lim_{t \to 2} (2t^3 + 8) = 2(2^3) + 8 = 16 + 8 = 24$
Upper bound: $\lim_{t \to 2} (t^4 + 8) = 2^4 + 8 = 16 + 8 = 24$

Step 2: Apply the Squeeze Theorem to find $\lim_{t \to 2} h(t)$ : Since both bounds equal $24$ , $\lim_{t \to 2} h(t) = 24$ .

Step 3: Evaluate the target expression using Limit Laws:

\lim_{t \to 2} \frac{h(t) + 5}{3} = \frac{\lim_{t \to 2} h(t) + 5}{3} = \frac{24 + 5}{3} = \frac{29}{3}

Connections & References

Parent Concepts: Limits, Solving Inequalities
Direct Theoretical Applications: Special Trigonometric Limits, Continuity
Course Links: Calc-1 Session 03 | Calc-1 Session 05