The Squeeze Theorem, also known as the Sandwich Theorem or Pinching Theorem, is a powerful tool for evaluating limits when direct substitution fails. It states that if we have three functions, $f(x)$, $g(x)$, and $h(x)$, such that $f(x) \leq g(x) \leq h(x)$ for all $x$ in an interval containing $c$ (except possibly at $c$ itself), and if $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} h(x) = L$, then $\lim_{x \to c} g(x) = L$. Essentially, if $g(x)$ is "squeezed" between two functions that approach the same limit, then $g(x)$ must also approach that limit.
In practice, this means you need to find two functions that bound the function you're trying to find the limit of. A common example involves trigonometric functions like sine and cosine, which are bounded between -1 and 1.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Squeeze Theorem | A. A value that a function approaches as the input approaches a certain value. |
| 2. Limit | B. A function that is greater than or equal to another function in a given interval. |
| 3. Bounded Function | C. A function whose values are restricted between an upper and lower limit. |
| 4. Upper Bound | D. If $f(x) \leq g(x) \leq h(x)$ and $\lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} g(x) = L$. |
| 5. Lower Bound | E. A function that is less than or equal to another function in a given interval. |
The Squeeze Theorem is useful when direct ______________ doesn't work. It states that if $f(x) \leq g(x) \leq h(x)$ near a point $c$, and the ___________ of $f(x)$ and $h(x)$ as $x$ approaches $c$ are equal to $L$, then the limit of $g(x)$ as $x$ approaches $c$ is also equal to _________. This is because $g(x)$ is "______________" between $f(x)$ and $h(x)$. A common application involves trigonometric functions that are __________ between -1 and 1.
Describe a scenario where using the Squeeze Theorem would be necessary to evaluate a limit. Provide an example function and the bounding functions you would use.
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