Sandwich Theorem

What is the Sandwich Theorem?

The Sandwich Theorem, or Squeeze Theorem, is a mathematical result used to evaluate limits. It states:

If three functions $f(x)$, $g(x)$, and $h(x)$ satisfy the inequality:

for all $x$ in some interval around $a$, except possibly at $a$, and if:

then:

In simpler terms, if $g(x)$ is "sandwiched" between two functions that converge to the same limit, then $g(x)$ must also converge to that limit.

Geometric Interpretation

The Sandwich Theorem can be visualized geometrically. If the graph of $g(x)$ lies between the graphs of $f(x)$ and $h(x)$ as $x$ approaches a point, and both $f(x)$ and $h(x)$ converge to the same value, the graph of $g(x)$ is "squeezed" to the same limit.

Example Problems

Example 1: Prove $\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0$.

Solution:

Step 1: Identify $f(x)$, $g(x)$, and $h(x)$.

From the properties of $\sin$, we know:

Multiplying by $x^2$ (a non-negative term) gives:

Here, $f(x) = -x^2$, $g(x) = x^2 \sin\left(\frac{1}{x}\right)$, and $h(x) = x^2$.

Step 2: Evaluate the limits of $f(x)$ and $h(x)$ as $x \to 0$:

Step 3: Apply the Sandwich Theorem:

Since $f(x) \leq g(x) \leq h(x)$ and both $f(x)$ and $h(x)$ approach $0$, it follows that:

Example 2: Prove $\lim_{x \to 0} x^2 \cos\left(\frac{1}{x}\right) = 0$.

Solution:

Step 1: Using the same logic as Example 1, we have:

Step 2: Limits of the bounding functions:

Step 3: Apply the Sandwich Theorem:

Applications of the Sandwich Theorem

• Proving limits for complex functions.
• Simplifying trigonometric and oscillatory limits.
• Establishing continuity and convergence in advanced calculus problems.

Practice Problem

Problem:

Prove that $\lim_{x \to 0} x^2 e^{-\frac{1}{x^2}} = 0$ using the Sandwich Theorem.

Hint:

Use the exponential property $e^{-x^2} \in (0, 1]$ for $x > 0$, and multiply by $x^2$ to establish bounds.