📚 Topic Summary
The Squeeze Theorem (also known as the Sandwich Theorem or the Pinching Theorem) is a powerful tool in calculus for finding the limit of a function. If we have three functions, $f(x)$, $g(x)$, and $h(x)$, such that $f(x) \le g(x) \le h(x)$ for all $x$ in an interval around a point $c$ (except possibly at $c$ itself), and if $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} h(x) = L$, then $\lim_{x \to c} g(x) = L$. In simpler terms, if $g(x)$ is 'squeezed' between two functions that approach the same limit, then $g(x)$ must also approach that limit.
This theorem is particularly useful when dealing with functions that are difficult to evaluate directly, especially those involving trigonometric functions like sine and cosine multiplied by functions that approach zero. By finding simpler bounding functions, we can determine the limit of the more complex function.
🔤 Part A: Vocabulary
Match the term with its correct definition:
- Term: Limit
- Term: Function
- Term: Inequality
- Term: Bounding Function
- Term: Squeeze Theorem
- Definition: A function that is either always greater than or equal to, or always less than or equal to, another function within a specified interval.
- Definition: A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
- Definition: A statement that compares two expressions using symbols like $\le$, $\ge$, <, or >.
- Definition: The value that a function approaches as the input approaches some value.
- Definition: If $f(x) \le g(x) \le h(x)$ when x is near a, and $\lim_{x \to a} f(x) = L = \lim_{x \to a} h(x)$, then $\lim_{x \to a} g(x) = L$.
(Match the numbers to the correct definitions)
✍️ Part B: Fill in the Blanks
Complete the following paragraph using the words provided below:
The Squeeze Theorem is helpful when finding the ________ of a function that is trapped between two other ________. If both bounding functions approach the same ________, then the function in the middle is ________ to approach that same limit. This technique is especially useful with ________ functions multiplied by expressions approaching zero.
Words: limit, forced, bounding functions, trigonometric, limit
🤔 Part C: Critical Thinking
Explain, in your own words, why the Squeeze Theorem is useful in evaluating limits that are otherwise difficult to compute directly. Provide an example of a type of function where the Squeeze Theorem is often applied, and why traditional limit evaluation techniques might fail for these functions.
