Limits and Continuity

Formal Definition of a Limit

The concept of a limit is fundamental in calculus, describing the behavior of a function as its input approaches a particular value. The formal definition uses precise language to ensure clarity and rigor.

• Definition: Let f(x) be defined on an open interval around c (except possibly at c itself). We say that the limit of f(x) as x approaches c is L, and write:

• Formal (Epsilon-Delta) Definition: For every number , there exists a corresponding number such that:

• This means that as x gets arbitrarily close to c (but not equal to c), f(x) gets arbitrarily close to L.

• Key Terms: Epsilon (\varepsilon) represents how close f(x) must be to L; Delta (\delta) represents how close x must be to c.

Properties and Examples

• Existence of Limit: The function must be defined on an open interval around c, except possibly at c itself.

• Example: Consider . The limit as x approaches any real number c is simply .

• Notation:

Graphical Interpretation

• On a graph, the value of f(x) as x approaches c from both sides should approach the same value L.

• If the left and right limits are equal, the limit exists.

Summary Table: Epsilon-Delta Definition

Additional info: The notes reference the formal definition of a limit, including epsilon-delta language, and provide an example with . The original content was fragmented, so academic context and structure were added for clarity.