1. **Stating the problem:** Sandwich's Theorem, also known as the Squeeze Theorem, helps find the limit of a function that is "squeezed" between two other functions whose limits are known and equal at a point. 2. **The theorem statement:** Suppose we have three functions $f(x)$, $g(x)$, and $h(x)$ such that for all $x$ near a point $a$ (except possibly at $a$ itself), $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$$ Then, $$\lim_{x \to a} f(x) = L$$ 3. **Explanation:** This means if $f(x)$ is always between $g(x)$ and $h(x)$ near $a$, and both $g(x)$ and $h(x)$ approach the same limit $L$ as $x$ approaches $a$, then $f(x)$ must also approach $L$. 4. **Important rules:** - The inequalities must hold near $a$ (except possibly at $a$). - The limits of the bounding functions must be equal. 5. **Example:** To find $$\lim_{x \to 0} x^2 \sin(\frac{1}{x})$$ we know $$-1 \leq \sin(\frac{1}{x}) \leq 1$$ Multiplying all parts by $x^2$ (which is always nonnegative), $$-x^2 \leq x^2 \sin(\frac{1}{x}) \leq x^2$$ Since $$\lim_{x \to 0} -x^2 = 0 = \lim_{x \to 0} x^2$$ by Sandwich's Theorem, $$\lim_{x \to 0} x^2 \sin(\frac{1}{x}) = 0$$ 6. **Summary:** Sandwich's Theorem is a powerful tool to find limits of complicated functions by "trapping" them between simpler functions with known limits.