Read limits and continuity from a graph

• Read left-hand and right-hand limits by tracing a graph toward the same input.
• Distinguish an approached y-value from the actual function value shown by an open or closed point.
• Apply the three conditions for continuity at a selected input.

Build the idea

Worked example

1. Step 1

   Following the graph from x-values less than two, the y-values move toward four even though the point at that height is open.

   $$\lim_{x\to2^-}f(x)=4$$

2. Step 2

   Following the graph from x-values greater than two also leads toward four, so the matching one-sided limits create a two-sided limit.

   $$\lim_{x\to2^+}f(x)=4$$

3. Step 3

   The filled point records f of two as one. Because one does not equal the limit four, the function fails the matching condition for continuity.

   $$f(2)=1\ne4$$

• Reading the filled point as the limit because it is the official function value.

  The limit follows nearby graph behavior. The filled point determines only f of the target input.

• Declaring a two-sided limit after inspecting only one side.

  Trace from both lower and higher x-values; the two approaches must agree before a two-sided limit exists.

A graph approaches negative two from the left of x equals three and five from the right. State both one-sided limits and the two-sided limit.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Lesson overview

   A limit describes the value a function approaches near an input. Continuity additionally requires the function to be defined there and to match that approached value. A limit asks what outputs settle toward as inputs move close to a target from both sides. The graph may have a hole at the target and still approach one common height. Read left-hand and right-hand limits by tracing a graph toward the same input.

2. Part 2

   Visual model and equations

   A two-sided limit exists only when the left-hand and right-hand approaches agree. If the graph heads toward different heights, no single two-sided value describes the nearby behavior. Trace a curve from x-values below and above two toward an open circle at height four, while a separate closed point at x equals two sits at height one. The limit of f of x as x approaches two equals four. For continuity at a, the limit as x approaches a must equal f of a.

   • The limit of f of x as x approaches two equals four.
   • For continuity at a, the limit as x approaches a must equal f of a.
3. Part 3

   Worked example

   A graph approaches height four from both sides of x equals two, has an open circle at two comma four, and a filled point at two comma one. Find the limit, the function value, and whether the function is continuous. Read each side's approaching height independently, read the filled point for the function value, then compare the values against the continuity conditions. Following the graph from x-values less than two, the y-values move toward four even though the point at that height is open. Following the graph from x-values greater than two also leads toward four, so the matching one-sided limits create a two-sided limit. The filled point records f of two as one. Because one does not equal the limit four, the function fails the matching condition for continuity. The limit is four, the function value is one, and the function is not continuous at x equals two. Both one-sided traces approach the same open-circle height, while the closed point is visibly elsewhere. This confirms that the limit exists but does not equal the defined value.

   • The left-hand limit of f of x as x approaches two equals four.
   • The right-hand limit of f of x as x approaches two equals four.
   • f of two equals one, which does not equal four.
4. Part 4

   Checks, practice, and scope

   Continuity is stricter than limit existence. The function value must exist, the two-sided limit must exist, and those two values must be equal. Reading the filled point as the limit because it is the official function value. The limit follows nearby graph behavior. The filled point determines only f of the target input. Declaring a two-sided limit after inspecting only one side. Trace from both lower and higher x-values; the two approaches must agree before a two-sided limit exists. A graph approaches negative two from the left of x equals three and five from the right. State both one-sided limits and the two-sided limit. Different one-sided values cannot combine into one two-sided limit. The left-hand limit is negative two, the right-hand limit is five, and the two-sided limit does not exist. Limits describe nearby approach behavior, not necessarily the value at the target. A two-sided limit requires matching left-hand and right-hand limits. Continuity requires definition, limit existence, and equality between the limit and function value. Graph reading provides qualitative or displayed exact values; numerical tables and algebraic limit laws require separate care. This lesson covers point continuity, not uniform continuity, endpoint conventions, or formal epsilon-delta proofs.

Scope and limitations

• Graph reading provides qualitative or displayed exact values; numerical tables and algebraic limit laws require separate care.
• This lesson covers point continuity, not uniform continuity, endpoint conventions, or formal epsilon-delta proofs.