1.2 Defining Limits and Using Limit Notation

• The formal definition of a limit involves epsilon and delta values.

  True
• What is a limit in calculus?

  Function's output as input approaches a value
• The notation for a limit as $x$ approaches $c$ is written as $\lim_{x \to c} f(x) =$$L$ True
• What does $L$ represent in limit notation?


  Value the function approaches
• When do infinite limits occur?

  Function grows without bound
• One-sided limits consider function behavior from the left and right
• What is the value of $\lim_{x \to c} f(x)$ in the case of an infinite limit?


  $\pm \infty$
• If the left and right limits differ, the limit does not exist
• What is the left-hand limit of f(x) = \begin{cases} x, & x < 2 \\ x^{2}, & x \geq 2 \end{cases}</latex> as $x \to 2^{ - }$?


  2
• Unbounded decay results in a limit of $- \infty$


  True
• A table of function values can be used to approximate a limit


  True
• What does evaluating limits numerically involve?

  Approximating with function values
• A limit describes how a function's output behaves as its input approaches a particular value.


  True
• In the limit notation $\lim_{x \to c} f(x) =$$L$, $L$ represents the value that the function $f(x)$ approaches as $x$ gets closer to c
• What does $\lim$ in the limit notation indicate?


  We are taking the limit
• The limit $\lim_{x \to 2} x^{2}$ is equal to 4.


  True
• Infinite limits indicate that the function grows or decays without bound as $x$ approaches $c$.


  True
• If the left-hand and right-hand limits are equal, the limit exists and is equal to their common value.


  True
• If the left and right-hand limits are not equal, the limit does not exist.
• To evaluate limits graphically, you must check if the function approaches the same value from both the left and right.

  True
• What kind of $x$ values should be considered when evaluating limits numerically?


  Close to but not equal to $c$
• Match the one-sided limit type with its definition:

  Left-Hand Limit ↔️ $\lim_{x \to c^{ - }} f(x) =$$L$

  Right-Hand Limit ↔️ $\lim_{x \to c^{ + }} f(x) =$$L$
• What is $\lim_{x \to 2} x^{2}$?


  4
• The informal definition of a limit states that as $x$ gets closer to $c$, $f(x)$ gets closer to $L$ True
• What does $\lim$ in limit notation indicate?


  Taking the limit
• The limit $\lim_{x \to 2} x^{2} =$$4$ is an example of a limit that exists.


  True
• What does $\lim_{x \to c^{ - }} f(x)$ denote?


  Left-hand limit
• What is \lim_{x \to 2^{+}} \frac{1}{x-2}</latex>?

  $\infty$
• The formal definition of a limit states that for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - c| < \delta$, then $|f(x) - L| < \epsilon$ True
• What does an infinite limit indicate as $x \to c$?


  Unbounded growth or decay
• Steps to evaluate limits graphically

  1️⃣ Analyze the graph of $f(x)$

  2️⃣ Check if the function approaches the same value from both left and right

  3️⃣ If left and right limits are equal, that value is the limit at $x =$$c$

  4️⃣ If they differ, the limit does not exist
• The left-hand limit as $x \to c$ is the value $f(x)$ approaches when $x < c$


  True
• The left-hand limit of $f(x)$ as $x \to 2^{ - }$ is 2
• What is the key principle for evaluating limits numerically?

  Consider values close to c
• What values of $x$ are considered when evaluating limits numerically?


  Close to but not equal to $c$
• As $x$ approaches $2$ in the function $f(x) =$$x^{2}$, the function values approach $4$.


  True
• What is the informal definition of a limit?

  As $x \to c$, f(x) \to L</latex>
• Order the following steps in evaluating the limit $\lim_{x \to 2} x^{2}$ numerically.


  1️⃣ Create a table of function values as $x$ approaches $2$

  2️⃣ Substitute values of $x$ close to $2$

  3️⃣ Observe the trend of function values

  4️⃣ Conclude that the limit is $4$
• What does $L$ represent in the limit notation $\lim_{x \to c} f(x) =$$L$?


  Value $f(x)$ approaches
• What does a one-sided limit describe?

  Function behavior from one side