Determining Limits Using the Squeeze Theorem - AP Calculus AB

1. Find $\lim_{x \to 0} x^2 \cos(\frac{1}{x})$ using the Squeeze Theorem.

   $0$. Since $-x^2 \leq x^2 \cos(\frac{1}{x}) \leq x^2$ and both bounds approach $0$.

2. Find $\lim_{x \to 0} x^2 \sin(\frac{1}{x})$ using the Squeeze Theorem.

   $0$. Since $-|x^2| \leq x^2 \sin(\frac{1}{x}) \leq |x^2|$ and both bounds approach $0$.

3. Determine $\lim_{x \to 0} x^4 \sin(\frac{1}{x})$ using the Squeeze Theorem.

   $0$. Since $-x^4 \leq x^4 \sin(\frac{1}{x}) \leq x^4$ and both bounds approach $0$.

4. What is the limit of $x \sin(\frac{1}{x})$ as $x \to 0$?

   $0$. Bounded between $-|x|$ and $|x|$, both approaching $0$.

5. What is $\lim_{x \to 0} x \sin(\frac{2}{x})$?

   $0$. Bounded between $-|x|$ and $|x|$, both approaching $0$.

6. Determine $\lim_{x \to 0} x^5 \sin(\frac{1}{x^6})$ using the Squeeze Theorem.

   $0$. Since $-x^5 \leq x^5 \sin(\frac{1}{x^6}) \leq x^5$ and both bounds approach $0$.

7. What is $\lim_{x \to 0} x \cos(\frac{1}{x^3})$?

   $0$. Bounded between $ -|x| $ and $ |x| $, both approaching $0$.

8. What must the limits of $g(x)$ and $h(x)$ equal for the Squeeze Theorem to apply?

   Both must equal $L$. Required for the theorem to conclude $f(x)$ has the same limit.

9. What is $\lim_{x \to 0} x^3 \sin(x^2)$ using the Squeeze Theorem?

   $0$. Since $|x^3 \sin(x^2)| \leq x^3$ and $x^3 \to 0$.

10. Find $\lim_{x \to 0} x^3 \sin(\frac{1}{x^5})$ using the Squeeze Theorem.

    $0$. Since $-x^3 \leq x^3 \sin(\frac{1}{x^5}) \leq x^3$ and both bounds approach $0$.

11. What inequality must $f(x)$ satisfy in the Squeeze Theorem?

    $g(x) \leq f(x) \leq h(x)$. This defines the sandwich relationship between the three functions.

12. What is the Squeeze Theorem used for in calculus?

    To determine the limit of a function. Forces a function between two bounds that approach the same limit.

13. What is $\lim_{x \to 0} x^4 \cos(\frac{1}{x^5})$?

    $0$. Since $-x^4 \leq x^4 \cos(\frac{1}{x^5}) \leq x^4$ and both bounds approach $0$.

14. Is $\lim_{x \to 0} x^2 \sin(\frac{1}{x})$ finite?

    Yes, it is $0$. The limit equals $0$, which is a finite value.

15. What is the limit of $x^2 \sin(\frac{1}{x^2})$ as $x \to 0$?

    $0$. Bounded between $-x^2$ and $x^2$, both approaching $0$.

16. What should $g(x)$ and $h(x)$ approach for $\lim_{x \to c}f(x)$ to exist?

    The same limit $L$. This ensures the squeezed function approaches the unique limit.

17. Is $\lim_{x \to 0} x^2 \sin(x^3)$ zero?

    Yes. Since $|x^2 \sin(x^3)| \leq x^2$ and $x^2 \to 0$.

18. Determine $\lim_{x \to 0} x^4 \cos(\frac{1}{x})$ using the Squeeze Theorem.

    $0$. Since $-x^4 \leq x^4 \cos(\frac{1}{x}) \leq x^4$ and both bounds approach $0$.

19. Find $\lim_{x \to 0} x^2 \sin(\frac{1}{x^3})$ using the Squeeze Theorem.

    $0$. Since $-x^2 \leq x^2 \sin(\frac{1}{x^3}) \leq x^2$ and both bounds approach $0$.

20. Determine $\lim_{x \to 0} x^3 \cos(\frac{1}{x^3})$ using the Squeeze Theorem.

    $0$. Since $-x^3 \leq x^3 \cos(\frac{1}{x^3}) \leq x^3$ and both bounds approach $0$.

21. What is $\lim_{x \to 0} x^2 \sin x$?

    $0$. Since $|x^2 \sin x| \leq x^2$ and $x^2 \to 0$.

22. Determine $\lim_{x \to 0} x^2 \cos(x^2)$ using the Squeeze Theorem.

    $0$. Since $|x^2 \cos(x^2)| \leq x^2$ and $x^2 \to 0$.

23. What is the limit $\lim_{x \to 0} x^2 \sin(\frac{1}{x^4})$?

    $0$. Since $-x^2 \leq x^2 \sin(\frac{1}{x^4}) \leq x^2$ and both bounds approach $0$.

24. Find $\lim_{x \to 0} x^3 \sin(\frac{1}{x})$ using the Squeeze Theorem.

    $0$. Since $-|x^3| \leq x^3 \sin(\frac{1}{x}) \leq |x^3|$ and both bounds approach $0$.

25. Determine $\lim_{x \to 0} x^3 \cos(\frac{1}{x^2})$ using the Squeeze Theorem.

    $0$. Since $-x^3 \leq x^3 \cos(\frac{1}{x^2}) \leq x^3$ and both bounds approach $0$.

26. Is $\lim_{x \to 0} x \sin x$ equal to zero?

    Yes. Since $|x \sin x| \leq |x|$ and $|x| \to 0$.

27. Find $\lim_{x \to 0} x^5 \sin(\frac{1}{x^4})$ using the Squeeze Theorem.

    $0$. Since $-x^5 \leq x^5 \sin(\frac{1}{x^4}) \leq x^5$ and both bounds approach $0$.

28. Find $\lim_{x \to 0} x^2 \cos x$ using the Squeeze Theorem.

    $0$. Since $|x^2 \cos x| \leq x^2$ and $x^2 \to 0$.

29. Find $\lim_{x \to 0} x \sin(\frac{1}{x^2})$ using the Squeeze Theorem.

    $0$. Since $-|x| \leq x \sin(\frac{1}{x^2}) \leq |x|$ and both bounds approach $0$.

30. Find $\lim_{x \to 0} x^3 \sin(\frac{2}{x})$ using the Squeeze Theorem.

    $0$. Since $-x^3 \leq x^3 \sin(\frac{2}{x}) \leq x^3$ and both bounds approach $0$.

31. Find $\lim_{x \to \infty} \frac{\sin x}{x}$ using the Squeeze Theorem.

    $0$. Since $-\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x}$ and both bounds approach $0$.

32. Determine $\lim_{x \to 0} x^3 \cos(x^4)$ using the Squeeze Theorem.

    $0$. Since $|x^3 \cos(x^4)| \leq x^3$ and $x^3 \to 0$.

33. What is the result of $\lim_{x \to 0} x \cos(\frac{1}{x})$?

    $0$. Bounded between $-|x|$ and $|x|$, both approaching $0$.

34. What is the limit $\lim_{x \to 0} x^2 \sin(x^2)$?

    $0$. Since $|x^2 \sin(x^2)| \leq x^2$ and $x^2 \to 0$.

35. State the Squeeze Theorem in terms of $f(x)$, $g(x)$, $h(x)$.

    If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to c}g(x) = \lim_{x \to c}h(x) = L$, then $\lim_{x \to c}f(x) = L$. The fundamental statement of the Squeeze Theorem with three functions.

36. Find $\lim_{x \to 0} x^2 \cos(\frac{2}{x})$ using the Squeeze Theorem.

    $0$. Since $-x^2 \leq x^2 \cos(\frac{2}{x}) \leq x^2$ and both bounds approach $0$.

37. What is $\lim_{x \to 0} x^4 \sin(\frac{1}{x^3})$?

    $0$. Since $-x^4 \leq x^4 \sin(\frac{1}{x^3}) \leq x^4$ and both bounds approach $0$.

38. Can the Squeeze Theorem be used if $\lim_{x \to c}g(x) \neq \lim_{x \to c}h(x)$?

    No. The bounding functions must approach the same value for the theorem to work.

39. Identify the conditions necessary for the Squeeze Theorem to apply.

    Functions $g(x)$, $f(x)$, $h(x)$ must satisfy $g(x) \leq f(x) \leq h(x)$ near $c$. The inequality must hold in a neighborhood around $c$.

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