1.8 Determining Limits Using the Squeeze Theorem

• The Squeeze Theorem states that if $f(x) ≤ g(x) ≤ h(x)$ and $\lim_{x \to c} f(x) =$$L$ and $\lim_{x \to c} h(x) =$$L$, then $\lim_{x \to c} g(x) =$$L$. This is useful when direct substitution
• One step in using the Squeeze Theorem is to identify bounding functions $f(x)$ and $h(x)$ such that f(x) ≤ g(x) ≤ h(x)</latex>.
• What is the definition of the lower bound function $f(x)$ in the Squeeze Theorem?


  $f(x) ≤ g(x)$
• In the example $\lim_{x \to 0} x^{2} \sin\left(\frac{1}{x}\right)$, the bounding functions are $- x^{2}$ and $x^{2}$ because $- 1 ≤ \sin\left(\frac{1}{x}\right) ≤ 1$ implies - x^{2} ≤ x^{2} \sin\left(\frac{1}{x}\right) ≤ x^{2}</latex>.
• What are the bounding functions for $\lim_{x \to 0} x \cos\left(\frac{1}{x}\right)$?


  $- |x| ≤ x \cos\left(\frac{1}{x}\right) ≤ |x|$
• Steps to use the Squeeze Theorem

  1️⃣ Find bounding functions $f(x)$ and $h(x)$ such that $f(x) ≤ g(x) ≤ h(x)$.

  2️⃣ Evaluate $\lim_{x \to c} f(x)$ and $\lim_{x \to c} h(x)$.

  3️⃣ Confirm that $\lim_{x \to c} f(x) =$$\lim_{x \to c} h(x) =$$L$.

  4️⃣ Conclude that $\lim_{x \to c} g(x) =$$L$.
• The Squeeze Theorem is also known as the Pinching Theorem.
• In the example $\lim_{x \to 0} x^{2} \sin\left(\frac{1}{x}\right)$, the bounding functions are $- x^{2}$ and $x^{2}$ because $- 1 ≤ \sin\left(\frac{1}{x}\right) ≤ 1$ implies - x^{2} ≤ x^{2} \sin\left(\frac{1}{x}\right) ≤ x^{2}</latex>.
• What are the bounding functions for $\lim_{x \to 0} x \cos\left(\frac{1}{x}\right)$?


  $- |x| ≤ x \cos\left(\frac{1}{x}\right) ≤ |x|$
• One condition for using the Squeeze Theorem is that the limits of the bounding functions must be equal.
• The first step in using the Squeeze Theorem is to identify bounding functions
• The Squeeze Theorem requires that the limits of the bounding functions are equal.
• Steps to find bounding functions for the Squeeze Theorem

  1️⃣ Analyze the target function $g(x)$

  2️⃣ Use trigonometric bounds such as $- 1 ≤ \sin(x) ≤ 1$ or $- 1 ≤ \cos(x) ≤ 1$

  3️⃣ Apply absolute values if needed
• A function $f(x)$ is a lower bound of $g(x)$ if $f(x) ≤ g(x)$ for all $x$ near $c$.
• What is the first step in evaluating the limits of bounding functions in the Squeeze Theorem?

  Determine the limit expression
• Ensuring that the limits of the bounding functions are equal is necessary to apply the Squeeze Theorem.
• What is another name for the Squeeze Theorem?

  Pinching Theorem
• The Squeeze Theorem can be used when direct substitution is not possible.
• The Squeeze Theorem requires identifying bounding functions
• The Squeeze Theorem is also known as the Pinching Theorem.
• The Squeeze Theorem requires that the target function is bounded between two other functions near a point $c$.
• To apply the Squeeze Theorem, you must identify two bounding functions.
• Steps to apply the Squeeze Theorem

  1️⃣ Identify bounding functions $f(x)$ and $h(x)$ such that $f(x) ≤ g(x) ≤ h(x)$ for $x$ near $c$.

  2️⃣ Show that $\lim_{x \to c} f(x) =$$L$ and $\lim_{x \to c} h(x) =$$L$.

  3️⃣ Conclude that $\lim_{x \to c} g(x) =$$L$.
• The limit of $x^{2} \sin\left(\frac{1}{x}\right)$ as $x \to 0$ is 0.
• The limit of $x \cos\left(\frac{1}{x}\right)$ as $x \to 0$ is 0.
• The trigonometric bounds $- 1 ≤ \sin(x) ≤ 1$ and $- 1 ≤ \cos(x) ≤ 1$ are useful for finding bounding functions.
• Trigonometric bounds can be used to find bounding functions when dealing with sine or cosine functions.
• What is a lower bound of a function $g(x)$?


  $f(x) ≤ g(x)$
• Steps to find bounding functions for a given function $g(x)$


  1️⃣ Analyze the target function $g(x)$ to identify potential bounding expressions

  2️⃣ Use trigonometric bounds such as $- 1 ≤ \sin(x) ≤ 1$ and $- 1 ≤ \cos(x) ≤ 1$

  3️⃣ Apply absolute values to ensure non-negative values if needed
• The trigonometric bound $- 1 ≤ \cos(x) ≤ 1$ is true for all $x$.
• For $g(x) =$$x^{2} \sin\left(\frac{1}{x}\right)$, what is the lower bound $f(x)$?


  $- x^{2}$
• Match the function with its definition or bound:

  Target function $g(x)$ ↔️ $g(x)$

  Lower bound $f(x)$ ↔️ $f(x) ≤ g(x)$

  Upper bound $h(x)$ ↔️ $g(x) ≤ h(x)$
• What condition must $f(x)$ satisfy to be a lower bound of $g(x)$?


  $f(x) ≤ g(x)$
• Steps to find bounding functions for a given function $g(x)$


  1️⃣ Analyze the target function $g(x)$ to identify potential bounding expressions

  2️⃣ Use trigonometric bounds such as $- 1 ≤ \sin(x) ≤ 1$ and $- 1 ≤ \cos(x) ≤ 1$

  3️⃣ Apply absolute values to ensure non-negative values if needed
• For $g(x) =$$x^{2} \sin\left(\frac{1}{x}\right)$, what is the upper bound $h(x)$?


  $x^{2}$
• Match the function with its definition or bound:

  Target function $g(x)$ ↔️ $g(x)$

  Lower bound $f(x)$ ↔️ $f(x) ≤ g(x)$

  Upper bound $h(x)$ ↔️ $g(x) ≤ h(x)$
• What is the first step in evaluating the limits of bounding functions when using the Squeeze Theorem?

  Determine the limit expression
• Steps to evaluate the limits of bounding functions for the Squeeze Theorem

  1️⃣ Determine the limit expression

  2️⃣ Substitute $x =$$c$

  3️⃣ Simplify

  4️⃣ Ensure equal limits
• If \lim_{x \to 1} (2x + 1) = 3</latex> and $\lim_{x \to 1} (3x - 1) =$$2$, the Squeeze Theorem can be applied.


  False
• For the Squeeze Theorem to be applicable, the bounding functions must have the same limit