1.8 Determining Limits Using the Squeeze Theorem

• The first step of the Squeeze Theorem is to identify a function f(x)
• The Squeeze Theorem requires that f(x) is bounded above by g(x)
• In the Squeeze Theorem, if h(x) ≤ f(x) ≤ g(x), then f(x) is said to be bounded
• The limits of g(x) and h(x) must approach the same value L as x approaches c.

  True
• If limx→c g(x) = limx→c h(x) = c^2, then limx→c f(x) = c^2.


  True
• The Squeeze Theorem can be applied to conclude that lim_x→c f(x) = c^2
• The function f(x) = x^2 is bounded above by g(x) = x^2 + 1
• Match the function with its corresponding limit as x→c:

  f(x) = x^2 ↔️ c^2

  g(x) = x^2 + 1 ↔️ c^2 + 1

  h(x) = x^2 - 1 ↔️ c^2 - 1
• What are the two conditions required for the Squeeze Theorem to apply?

  Bounded f(x) and equal limits
• In the Squeeze Theorem, f(x) must be bounded above by g(x) and bounded below by h(x) for all x in an interval around c
• Match the function with its condition and limit as x→c:

  f(x) = x² ↔️ Bounded above by g(x) = x² + 1 and below by h(x) = x² - 1 ||| lim_x→c f(x) = c²

  g(x) = x² + 1 ↔️ - ||| lim_x→c g(x) = c² + 1

  h(x) = x² - 1 ↔️ - ||| lim_x→c h(x) = c² - 1
• The limits of g(x) and h(x) must be equal as x approaches c to use the Squeeze Theorem.


  True
• What must be shown about the limits of g(x) and h(x) as x approaches a value c in order to apply the Squeeze Theorem?


  They are equal
• What is the upper bound for f(x) = x^2 in the given example?


  x^2 + 1</latex>
• If **lim**_x→c g(x) = lim_x→c h(x), then lim_x→c f(x) must also equal that value according to the Squeeze Theorem.


  True
• For the Squeeze Theorem to apply, both bounding functions g(x) and h(x) must approach the same limit
• The Squeeze Theorem requires that f(x) is bounded between two other functions.

  True
• What does the Squeeze Theorem state about a function f(x) bounded above by g(x) and below by h(x)?

  Its limit equals g(x)'s and h(x)'s
• The Squeeze Theorem requires that h(x) ≤ f(x) ≤ g(x)
• In the example, f(x) = x^2 is bounded above by g(x) = x^2 + 1
• The limits of the bounding functions g(x) and h(x) must be equal
• What must the limit of f(x) be equal to if the limits of g(x) and h(x) are equal as x approaches c?

  c
• What is the limit of f(x) = x^2 as x approaches c, according to the Squeeze Theorem?

  c^2
• For the Squeeze Theorem to apply, the function f(x) must be bounded between two other functions.

  True
• One condition of the Squeeze Theorem is that f(x) must be bounded
• What happens to the limit of f(x) as x approaches c if the conditions of the Squeeze Theorem are satisfied?

  **lim**_x→c **f(x)** = **L**
• The functions g(x) and h(x) must have the same limit as x approaches c for the Squeeze Theorem to apply.


  True
• In using the Squeeze Theorem, we must identify a function f(x) that is bounded above by g(x) and bounded below by h(x).bounded
• To use the Squeeze Theorem, we need to identify a function f(x) that is bounded above by g(x) and bounded below by h(x)
• What is the lower bound for f(x) = x^2 in the given example?


  $x^{2} - 1$
• What is the common limit of the upper and lower bounds for f(x) = x^2 as x approaches c?


  $c^{2}$
• What is the primary condition for the Squeeze Theorem regarding the function f(x)?


  It is bounded above
• As x approaches c, both g(x) and h(x) must approach the same limit
• The limits of g(x) and h(x) must be equal as x approaches c for the Squeeze Theorem to apply.


  True
• The function g(x) = x^2 + 1 has a limit of c^2 + 1 as x approaches c.


  True
• What is the limit of f(x) = x^2 as x approaches c if it is bounded by g(x) = x^2 + 1 and h(x) = x^2 - 1?

  c^2
• What condition is required for the Squeeze Theorem to be applied to a function f(x)?

  Bounded above and below
• Match the function with its limit as x approaches c:

  f(x) = x^2 ↔️ c^2

  g(x) = x^2 + 1 ↔️ c^2 + 1

  h(x) = x^2 - 1 ↔️ c^2 - 1
• Steps to show that upper and lower bounds approach the same limit:

  1️⃣ Demonstrate that the limits of g(x) and h(x) are equal

  2️⃣ Apply the Squeeze Theorem

  3️⃣ Conclude that the limit of f(x) is the same
• The Squeeze Theorem allows us to conclude that the limit of f(x) as x approaches c is equal to the common limit of its bounds