Real-world applications and examples of the Squeeze Theorem

📚 Quick Study Guide

• 🔢 The Squeeze Theorem states that if $g(x) \leq f(x) \leq h(x)$ for all $x$ in an interval containing $c$ (except possibly at $c$ itself), and if $\lim_{x \to c} g(x) = L = \lim_{x \to c} h(x)$, then $\lim_{x \to c} f(x) = L$.
• 💡 This is particularly useful when dealing with functions that oscillate or are otherwise difficult to evaluate directly.
• 📐 Key applications include finding limits of trigonometric functions, especially those involving $\sin$ and $\cos$ multiplied by functions that approach zero.
• 🧪 Real-world examples often involve error bounds in approximations or bounding complex systems within simpler, solvable models.
• 📝 Remember to always check that the bounding functions $g(x)$ and $h(x)$ have the same limit!

Practice Quiz

1. Question 1: Which of the following is a typical scenario where the Squeeze Theorem is most useful?
   1. A) Finding the derivative of a polynomial.
   2. B) Evaluating the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0.
   3. C) Integrating an exponential function.
   4. D) Solving a linear equation.
2. Question 2: Suppose $2x \leq f(x) \leq x^2 + 1$ for all $x$ near 1. What is $\lim_{x \to 1} f(x)$?
   1. A) 1
   2. B) 2
   3. C) 3
   4. D) The limit cannot be determined.
3. Question 3: In the context of error estimation, how might the Squeeze Theorem be applied?
   1. A) To find the exact error in a measurement.
   2. B) To bound the error between two known values.
   3. C) To eliminate error entirely.
   4. D) To amplify the error for better detection.
4. Question 4: Which condition MUST be satisfied to apply the Squeeze Theorem?
   1. A) $f(x)$ must be continuous.
   2. B) $g(x) \leq f(x) \leq h(x)$ must hold.
   3. C) $g(x)$ and $h(x)$ must be polynomials.
   4. D) $f(x)$ must be differentiable.
5. Question 5: Consider $f(x) = x^2 \cos(\frac{1}{x})$. What functions could be used to 'squeeze' $f(x)$ as $x$ approaches 0?
   1. A) $-x^2$ and $x^2$
   2. B) $-\cos(\frac{1}{x})$ and $\cos(\frac{1}{x})$
   3. C) $-x$ and $x$
   4. D) $-1$ and $1$
6. Question 6: How is the Squeeze Theorem used in proving that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$?
   1. A) It is not used; L'Hopital's Rule is used instead.
   2. B) It is used to bound $\frac{\sin(x)}{x}$ between two functions that both approach 1.
   3. C) It is used to directly compute the value of $\sin(0)$.
   4. D) It is used to simplify the expression before taking the limit.
7. Question 7: What is the limit of $x^4 \sin(\frac{1}{x^2})$ as $x$ approaches 0, using the Squeeze Theorem?
   1. A) 0
   2. B) 1
   3. C) ∞
   4. D) Does not exist.