The Squeeze Theorem, also known as the Sandwich Theorem or Pinching Theorem, is a fundamental result in calculus that helps in finding the limit of a function. It is especially useful when the function of interest is difficult to evaluate directly, but can be "squeezed" between two other functions whose limits are easier to compute.

Squeeze Theorem Statement:

Let $f(x)$, $g(x)$, and $h(x)$ be functions defined on some interval around a point $c$ (except possibly at $c$ itself). If:

$g(x) \leq f(x) \leq h(x)$

for all $x$ in this interval, and:

$\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$

then:

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

How It Works:

• Functions $g(x)$ and $h(x)$: These are the "bounding" functions, which "squeeze" $f(x)$ from below and above.
• Function $f(x)$: This is the function whose limit you're trying to determine.
• Point $c$: The point at which you're evaluating the limit.
• Limit $L$: The common limit of $g(x)$ and $h(x)$ at point $c$.

Example:

Suppose we want to find the limit of $f(x) = x^2 \sin\left(\frac{1}{x}\right)$ as $x$ approaches 0.

1. Bounding Functions: We know that $-1 \leq \sin\left(\frac{1}{x}\right) \leq 1$.
2. Multiply by $x^2$: This gives $-x^2 \leq x^2 \sin\left(\frac{1}{x}\right) \leq x^2$.
3. Limits of the Bounding Functions: As $x$ approaches 0, both $-x^2$ and $x^2$ approach 0.

Using the Squeeze Theorem, we conclude:

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

Would you like further details or have any questions?

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Here are 5 related questions:

1. Can the Squeeze Theorem be applied to sequences?
2. What are some common functions where the Squeeze Theorem is frequently used?
3. How is the Squeeze Theorem used in proving the limit of trigonometric functions?
4. Can the Squeeze Theorem be applied in higher dimensions (multivariable calculus)?
5. What are some limitations of the Squeeze Theorem?

Tip: The Squeeze Theorem is particularly useful when dealing with oscillating functions like sine or cosine, especially when multiplied by a function that tends to zero.