The Sandwich Theorem

Definition

The Sandwich Theorem (or Squeeze Theorem) states that if a function $f(x)$ is “sandwiched” between two other functions $g(x)$ and $h(x)$, and both $g$ and $h$ approach the same limit $L$ at a point $c$, then $f$ must also approach $L$ at that point. If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$.

• How to read: “The g of x is less than or equal to f of x is less than or equal to h of x; if the limits of g and h as x approaches c both equal L, then the limit of f as x approaches c equals L.”
• Meaning: A trapped function cannot escape the squeeze—if both bounds converge to $L$, the middle must too.

Why It Matters

The Sandwich Theorem is the ‘squeeze’ that solves the unsolvable; it allows mathematicians and physicists to pin down the behavior of wild, oscillating functions by trapping them between two predictable boundaries.

Core Concepts

• Indirect Proof: This theorem is used when a function is too “wild” or complex to evaluate directly, but is bounded by simpler functions.
• Boundary Control: You don’t need to know exactly what $f(x)$ is doing; you only need to know that it is “trapped” by its boundaries.
• Oscillation Dampening: It is the primary tool for evaluating limits of oscillating functions like $x^2 \sin(1/x)$ as $x \to 0$.

• How to read: “The x squared sine of one divided by x as x approaches zero.”
  • Meaning / when to use: Sine oscillates wildly, but $x^2 \to 0$ crushes the amplitude—bound between $-x^2$ and $x^2$ to get limit $0$.

Connected Concepts

• The Binomial Theorem
• Mean Value Theorem for Integrals
• Comparison Theorem for Improper Integrals