The Squeeze Theorem (or Sandwich Theorem) is a fundamental tool in calculus used to evaluate limits of functions. It is particularly useful when a function's behavior is difficult to analyze directly, but it can be "squeezed" between two simpler functions whose limits are known and equal at a specific point.

1. What is the Squeeze Theorem?

The Squeeze Theorem states:

If g(x)≤f(x)≤h(x)g(x) \leq f(x) \leq h(x)g(x)≤f(x)≤h(x) for all xxx in an interval (a−δ,a+δ)(a - \delta, a + \delta)(a−δ,a+δ), except possibly at x=ax = ax=a, and if:

lim⁡x→ag(x)=lim⁡x→ah(x)=L,\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L,x→alim​g(x)=x→alim​h(x)=L,

then:

lim⁡x→af(x)=L.\lim_{x \to a} f(x) = L.x→alim​f(x)=L.

This can also be applied to limits as x→∞x \to \inftyx→∞ or x→−∞x \to -\inftyx→−∞.

2. When to Use the Squeeze Theorem

You should consider using the Squeeze Theorem in the following scenarios:

a) Bounding a Difficult Function

• When f(x)f(x)f(x) is difficult to evaluate directly, but you can find two simpler functions, g(x)g(x)g(x) and h(x)h(x)h(x), such that g(x)≤f(x)≤h(x)g(x) \leq f(x) \leq h(x)g(x)≤f(x)≤h(x).

b) Oscillatory Functions

• Functions like f(x)=x2sin⁡(1x)f(x) = x^2 \sin\left(\frac{1}{x}\right)f(x)=x2sin(x1​), where direct evaluation is challenging due to oscillatory behavior.

c) Behavior Near Discontinuities

• When analyzing functions near points where they are undefined or behave irregularly.

d) Small Perturbations

• When a small additive or multiplicative term complicates direct computation, but its effect diminishes as x→ax \to ax→a or x→∞x \to \inftyx→∞.

3. Steps to Apply the Squeeze Theorem

Step 1: Find Bounding Functions

Identify two functions g(x)g(x)g(x) and h(x)h(x)h(x) such that:

g(x)≤f(x)≤h(x).g(x) \leq f(x) \leq h(x).g(x)≤f(x)≤h(x).

Step 2: Check Validity of Inequalities

Ensure that g(x)≤f(x)≤h(x)g(x) \leq f(x) \leq h(x)g(x)≤f(x)≤h(x) holds for xxx in a neighborhood around aaa, excluding possibly aaa itself.

Step 3: Calculate the Limits of the Bounds

Compute:

lim⁡x→ag(x)=Landlim⁡x→ah(x)=L.\lim_{x \to a} g(x) = L \quad \text{and} \quad \lim_{x \to a} h(x) = L.x→alim​g(x)=Landx→alim​h(x)=L.

Step 4: Conclude Using the Theorem

If both bounding limits are equal (LLL), then by the Squeeze Theorem:

lim⁡x→af(x)=L.\lim_{x \to a} f(x) = L.x→alim​f(x)=L.

4. Examples

Example 1: A Trigonometric Limit

Evaluate:

lim⁡x→0x2sin⁡(1x).\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right).x→0lim​x2sin(x1​).

• Step 1: Identify bounds. Since −1≤sin⁡(1x)≤1-1 \leq \sin\left(\frac{1}{x}\right) \leq 1−1≤sin(x1​)≤1:

−x2≤x2sin⁡(1x)≤x2.-x^2 \leq x^2 \sin\left(\frac{1}{x}\right) \leq x^2.−x2≤x2sin(x1​)≤x2.

• Step 2: Compute limits of bounds as x→0x \to 0x→0:

lim⁡x→0−x2=0andlim⁡x→0x2=0.\lim_{x \to 0} -x^2 = 0 \quad \text{and} \quad \lim_{x \to 0} x^2 = 0.x→0lim​−x2=0andx→0lim​x2=0.

• Step 3: Apply the Squeeze Theorem:

lim⁡x→0x2sin⁡(1x)=0.\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0.x→0lim​x2sin(x1​)=0.

Example 2: A Rational Function

Evaluate:

lim⁡x→∞sin⁡(x)x.\lim_{x \to \infty} \frac{\sin(x)}{x}.x→∞lim​xsin(x)​.

• Step 1: Identify bounds. Since −1≤sin⁡(x)≤1-1 \leq \sin(x) \leq 1−1≤sin(x)≤1:

−1x≤sin⁡(x)x≤1x.-\frac{1}{x} \leq \frac{\sin(x)}{x} \leq \frac{1}{x}.−x1​≤xsin(x)​≤x1​.

• Step 2: Compute limits of bounds as x→∞x \to \inftyx→∞:

lim⁡x→∞−1x=0andlim⁡x→∞1x=0.\lim_{x \to \infty} -\frac{1}{x} = 0 \quad \text{and} \quad \lim_{x \to \infty} \frac{1}{x} = 0.x→∞lim​−x1​=0andx→∞lim​x1​=0.

• Step 3: Apply the Squeeze Theorem:

lim⁡x→∞sin⁡(x)x=0.\lim_{x \to \infty} \frac{\sin(x)}{x} = 0.x→∞lim​xsin(x)​=0.

5. Common Mistakes

1. Failing to Establish Valid Bounds
   Ensure g(x)≤f(x)≤h(x)g(x) \leq f(x) \leq h(x)g(x)≤f(x)≤h(x) is true for all xxx in the specified interval.

2. Mismatch of Limits
   Both g(x)g(x)g(x) and h(x)h(x)h(x) must converge to the same limit.

3. Misinterpreting Oscillation
   Understand that bounded oscillatory behavior can often be squeezed.

6. Why the Squeeze Theorem Works

The intuition behind the theorem lies in the behavior of limits. If f(x)f(x)f(x) is always trapped between g(x)g(x)g(x) and h(x)h(x)h(x), and both g(x)g(x)g(x) and h(x)h(x)h(x) approach the same value, f(x)f(x)f(x) is compelled to approach that value too. This is particularly useful when f(x)f(x)f(x) oscillates or behaves erratically.

7. Applications of the Squeeze Theorem

• Physics: Analyzing small oscillations or limits in wave equations.

• Engineering: Studying systems with bounded perturbations.

• Mathematics: Proving limit-related theorems and establishing convergence.

8. Summary

The Squeeze Theorem is a powerful technique for evaluating tricky limits. To use it effectively:

• Find appropriate bounding functions.

• Ensure the bounds and limits align correctly.

• Use the theorem when direct computation is infeasible due to oscillations, irregular behavior, or complex expressions.

By mastering this theorem, you gain a versatile tool for tackling challenging problems in calculus!