The Squeeze Theorem (also known as the Sandwich Theorem) states that if a function $f(x)$ is "squeezed" between two other functions $g(x)$ and $h(x)$ near a point $c$, and the limits of these two functions are equal as $x$ approaches $c$, then the limit of $f(x)$ at $c$ must also exist and equal that same value.

Formally: $\text{If } g(x) \leq f(x) \leq h(x) \quad \text{for all } x \text{ near } c \quad \text{and} \quad \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L,$ $\text{then} \quad \lim_{x \to c} f(x) = L.$

Would you like a detailed explanation or a few solved problems? In the meantime, here are five practice questions related to the squeeze theorem.

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5 Practice Questions:

1. Basic limit example: Use the squeeze theorem to evaluate $\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right)$.
2. Oscillating function: Show that $\lim_{x \to \infty} \frac{\sin x}{x} = 0$ using the squeeze theorem.
3. Absolute value bound: Prove that $\lim_{x \to 0} x^2 \cos\left(\frac{1}{x}\right) = 0$.
4. Trig limit: Use the squeeze theorem to find $\lim_{\theta \to 0} \theta \sin\left(\frac{1}{\theta}\right)$.
5. Non-trig function: Evaluate $\lim_{x \to \infty} \frac{\cos x}{x^2}$ using the squeeze theorem.

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Tip:

When using the Squeeze Theorem, look for functions that oscillate (e.g., $\sin$, $\cos$), and then find simple upper and lower bounds around them.