1.2 Defining Limits and Using Limit Notation

• What is a limit in mathematical terms?

  A value a function approaches
• The limit $\lim_{x \to a} f(x) =$$L$ means that as $x$ approaches $a$, the function $f(x)$ approaches L
• What is the limit of $x^{2}$ as $x$ approaches 2?


  4
• The ε-δ definition provides a formal way to define the limit of a function.
• In the ε-δ definition, if 0 < |x - a| < δ</latex>, then $|f(x) - L| < ε$, where $ε$ represents the desired level of closeness
• In the ε-δ proof for $\lim_{x \to 2} (2x + 1) =$$5$, what value of $δ$ should be chosen?


  $δ =$$ε / 2$
• Match the definition aspect with its corresponding type:

  Less precise ↔️ Informal definition

  Highly precise ↔️ ε-δ definition
• The ε-δ definition is commonly used in early calculus.

  False
• The general notation for a limit is \lim_{x \to a} f(x) = L</latex>, where $L$ represents the limit value
• What does $\lim_{x \to 3} (x^{2} +$$1) =$$10$ mean in words?


  As $x$ approaches 3, $x^{2} +$$1$ approaches 10
• Steps to define a limit using the ε-δ definition

  1️⃣ For every $ε > 0$, find a $δ > 0$

  2️⃣ Whenever $0 < |x - a| < δ$, show that $|f(x) - L| < ε$

  3️⃣ Confirm the limit $\lim_{x \to a} f(x) =$$L$
• $\lim_{x \to 2} x^{2} =$$4$ indicates that $x^{2}$ approaches $4$ as $x$ gets closer to $2$.
• A limit describes the value a function approaches as its input gets arbitrarily close to a specific point
• The expression $\lim_{x \to a} f(x) =$$L$ means that as $x$ approaches $a$, $f(x)$ approaches $L$.
• What does $x \to a$ mean in limit notation?


  $x$ approaches $a$
• As $x$ approaches $2$, the limit of $x^{2}$ is 4
• The limit of $x^{2}$ as $x$ approaches $2$ is 4</latex>.
• The ε-δ definition states that $\lim_{x \to a} f(x) =$$L$ if for every $ε > 0$, there exists a $δ > 0$ such that whenever $0 < |x - a| < δ$, we have $|f(x) - L| < ε$.L
• The ε-δ definition ensures that $f(x)$ is within $ε$ of $L$ for all $x$ within $δ$ of $a$.
• How is $δ$ chosen for $f(x) =$$2x +$$1$ to verify $\lim_{x \to 2} f(x) =$$5$?


  $δ =$$ε / 2$
• Match the definition aspect with its type:

  Precision ↔️ Highly precise for ε-δ definition

  Context ↔️ Early calculus for informal definition
• The general notation for a limit is $\lim_{x \to a} f(x) =$$L$, where $L$ represents the limit value
• $x \to a$ indicates that $x$ is approaching $a$ in limit notation.
• What does $\lim_{x \to 3} (x^{2} +$$1) =$$10$ mean in plain language?


  As $x$ approaches $3$, $x^{2} +$$1$ approaches $10$
• Limit laws allow us to evaluate complex limits by breaking them down into simpler components
• $\lim_{x \to a} [f(x) +$$g(x)] =$$\lim_{x \to a} f(x) +$$\lim_{x \to a} g(x)$ is an example of the addition limit law.
• What condition must be satisfied for the division limit law to be valid?

  \lim_{x \to a} g(x) \neq0</latex>
• Order the steps to evaluate $\lim_{x \to 4} (x^{2})^{2}$ using limit laws:


  1️⃣ $\lim_{x \to 4} (x^{2})^{2} =$$[\lim_{x \to 4} x^{2}]^{2}$

  2️⃣ $[\lim_{x \to 4} x^{2}]^{2} =$$(16)^{2}$

  3️⃣ $(16)^{2} =$$256$
• What is the constant multiple property for limits?

  $\lim_{x \to a} [c \cdot f(x)] =$$c \cdot \lim_{x \to a} f(x)$
• What is the power property for limits?

  \lim_{x \to a} [f(x)]^{n} = [\lim_{x \to a} f(x)]^{n}</latex>
• What is the root property for limits?

  $\lim_{x \to a} \sqrt[n]{f(x)} =$$\sqrt[n]{\lim_{x \to a} f(x)}$
• The addition property for limits states that \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)</latex>. This simplifies the evaluation of complex limits
• The subtraction property for limits states that $\lim_{x \to a} [f(x) - g(x)] =$$\lim_{x \to a} f(x) - \lim_{x \to a} g(x)$.
• What is the multiplication property for limits?

  \lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)</latex>
• The division property for limits states that $\lim_{x \to a} \frac{f(x)}{g(x)} =$$\frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided that \lim_{x \to a} g(x) \neq 0
• A limit exists if the left-hand limit and the right-hand limit are equal.
• What does the left-hand limit refer to in the context of calculating limits graphically?

  Value $f(x)$ approaches as $x$ approaches $a$ from the left ($x < a$)
• What does the right-hand limit refer to in the context of calculating limits graphically?

  Value $f(x)$ approaches as $x$ approaches $a$ from the right ($x > a$)
• What is a limit in the context of a function's behavior as it approaches a specific point on its graph?

  The value it approaches
• If $f(x)$ approaches $3$ as $x$ approaches $2$, we can write $\lim_{x \to 2} f(x) =$$3$.