Mathematical Limits and Continuity

Variable Definitions: In the context of the limit notation, $c$ represents the specific $x$ -values being approached, while the limit itself is the corresponding $y$ -value.
Directional Limits: Limits are evaluated by observing the behavior of the function from both the left-hand side and the right-hand side of a given point $c$ .
The Proximity Principle: Limits only care about the behavior of the function as it gets extremely close to a specific $x$ -value. The actual value of the function at that point is irrelevant to the existence or value of the limit.
- Example: For a specific function, as $x$ approaches $2$ , the $y$ -values from both directions may approach $1$ . Even if the function value at $x = 2$ is undefined or exists elsewhere, the limit as $x$ approaches $2$ is $1$ .

Undefined Points: A limit can exist even if the function is undefined at $x = c$ . If the curves from the left and right meet at the same location on a graph, that location is the limit, even if there is a hole at that exact spot.
Mismatching Values: A function may be defined at $x = c$ , but that functional value might not equal the limit.
- Case Study: As $x$ approaches $2$ , the limit of the function is $1$ . However, the functional value, denoted as $f(2)$ , is represented by a separate point at $y = 3$ . In this scenario, the limit is $1$ while the functional value is $3$ .
Matching Values: When a point exists directly on the path of the curve without any breaks or separate points, the limit value and functional value are the same.
- Example: At the point $(1, 1)$ , when the notation asks for the functional value when $x = 1$ , the answer is $1$ . This indicates that at $x = 1$ , there is a point exactly at $y = 1$ .
Holes and Approximations: At $x = 2$ , if the graph from both sides approaches the same hole, they are approaching the same location on the "wall." If that hole is at $y = 0$ , then the limit is $0$ , even if the functional value exists elsewhere (e.g., at $y = 3$ ).

The Pen-Tracing Test: A practical method for identifying discontinuities is to trace the graph with a pen. Any point where you must pick up the pen to continue to the next part of the graph is considered a point of discontinuity.
Types of Discontinuity:
- Jump Discontinuity: Occurs when the graph "jumps" from one $y$ -value to another at a specific $x$ -value. For example, at $x = 0$ , the graph comes from different directions and does not meet, requiring a jump to continue tracing.
Specific Locations of Discontinuity: Based on the observed graph, discontinuities appear at the following $x$ -values:
- $x = 0$
- $x = 1$
- $x = 2$
- $x = 4$

Limit Existence: For a limit to exist at a point $c$ , the one-sided limits from the negative side (left) and the positive side (right) must be equal.
- $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$
Continuity Criteria: To determine if a function is continuous at a specific point, three conditions must be met:
1. Existence of the Limit: The limit as $x$ approaches $c$ must exist ( $\lim_{x \to c} f(x) = L$ ).
2. Existence of the Function: The function must be defined at $c$ (the functional value $f(c)$ must exist).
3. Equality: The limit value must equal the functional value ( $\lim_{x \to c} f(x) = f(c)$ ).

Question: For a discontinuity at zero where the graph is coming from different directions from the left and right, what would the limit be at zero?
Response: This refers to a jump discontinuity. If the left-hand and right-hand limits are not equal, the limit does not exist.
Question: For a limit to actually exist, do the negative and positive sides have to be equal, and does $y$ also have to equal that same number?
Response: That is the distinction between a limit existing and a function being continuous. For the limit alone to exist, the left and right approaches must meet at the same $y$ -value. For continuity at that point, you must also check if the function exists at that point and if the function value matches the limit value.