1.3 Estimating Limit Values from Graphs

• The right-hand limit is denoted by $\lim_{x \to c^{ + }} f(x)$.
• In calculus, what does a limit describe?

  Value a function approaches
• The formal notation for a limit is \lim_{x \to c} f(x) = L
• A function is continuous at a point if the limit at that point equals the function's value.
• Match the condition for continuity with its description:

  f(c) exists ↔️ Function is defined at c

  \lim_{x \to c} f(x) exists ↔️ Limit approaches a finite value

  \lim_{x \to c} f(x) = f(c) ↔️ Limit equals function's value at c
• Continuity at a point $c$ requires that $f(c)$ exists
• What is the formal notation for a limit?

  $\lim_{x \to c} f(x) =$$L$
• Continuity at a point requires that the function is defined, the limit exists, and the limit equals the function's value.
• Steps to estimate a limit value from a graph

  1️⃣ Identify the value $c$ that $x$ approaches.

  2️⃣ Examine the graph as $x$ approaches $c$ from the left.

  3️⃣ Examine the graph as $x$ approaches $c$ from the right.

  4️⃣ Estimate the value $L$ that $f(x)$ approaches.
• A limit describes the value a function approaches as its input gets arbitrarily close to a specific value
• For the function $f(x) =$$x +$$1$, what value does f(x)</latex> approach as $x$ approaches $2$?


  3
• Estimating limits from a graph involves checking both left-hand and right-hand limits.
• When estimating a limit from a graph, the first step is to identify the input value
• Steps to estimate a limit from a graph

  1️⃣ Identify the input value

  2️⃣ Look at the graph near that input value

  3️⃣ Check the left-hand limit

  4️⃣ Check the right-hand limit

  5️⃣ Compare both limits
• What is the left-hand limit of $f(x)$ as $x$ approaches 3</latex> in the example given?


  2
• What is the right-hand limit of $f(x)$ as $x$ approaches $3$ in the example given?


  2
• The left-hand and right-hand limits must be equal for the overall limit to exist.
• One-sided limits are directional limits that consider only one side of a specific input value
• Match the approach to $x$ with its limit notation:


  Left approach to $2$ ↔️ $\lim_{x \to 2^{ - }} f(x)$

  Right approach to $2$ ↔️ $\lim_{x \to 2^{ + }} f(x)$
• What is the left-hand limit of $f(x)$ as $x$ approaches $2$ in the second example?


  3
• What is the right-hand limit of $f(x)$ as $x$ approaches $2$ in the second example?


  5
• The left-hand limit is the value a function approaches as $x$ approaches $c$ from the left.
• One-sided limits are denoted as $\lim_{x \to c^{ - }} f(x)$ for the left-hand limit and \lim_{x \to c^{ + }} f(x)</latex> for the right-hand limit
• What is the notation for the left-hand limit as $x$ approaches $c$?


  $\lim_{x \to c^{ - }} f(x)$
• What is the notation for the right-hand limit as $x$ approaches $c$?


  $\lim_{x \to c^{ + }} f(x)$
• If both the left-hand and right-hand limits of a function at a point are equal, the overall limit exists at that point.
• The left-hand limit is the value the function approaches as $x$ approaches $c$ from the left
• What is the left-hand limit of a function as $x$ approaches $c$?


  Value from the left
• The right-hand limit of a function is the value the function approaches as $x$ approaches $c$ from the right
• If the left-hand limit and right-hand limit of a function at a point are equal, the overall limit exists at that point.
• For continuity at a point $c$, the limit as $x$ approaches $c$ must exist and be equal to f(c)
• If a function is continuous at a point $c$, then $\lim_{x \to c} f(x) =$$f(c)$.
• What does a limit graphically represent?

  Value function approaches
• Steps to estimate a limit value from a graph

  1️⃣ Identify the input value where the limit is being taken

  2️⃣ Look at the graph near that input value

  3️⃣ Check the left-hand limit

  4️⃣ Check the right-hand limit

  5️⃣ Compare both limits
• If the left-hand limit and right-hand limit of a function at a point differ, the limit does not exist.
• If the left-hand limit and the right-hand limit are equal, that value is the limit
• If the left-hand and right-hand limits of a function at a point differ, the limit exists at that point.

  False
• Steps to estimate the value of a limit from a graph

  1️⃣ Identify the input value where the limit is being taken

  2️⃣ Look at the graph near that input value

  3️⃣ Check the left-hand limit

  4️⃣ Check the right-hand limit

  5️⃣ Compare both limits
• To check the left-hand limit, we examine the value of $f(x)$ as $x$ approaches the input value from the left.
• If the left-hand limit and the right-hand limit are different, the limit does not exist.