Mathematics 1 - Limits and Continuity

Limits:

• A limit describes the behavior of a function as its input approaches a certain value.
• The limit of a function f(x) as x approaches a is denoted by lim(x→a) f(x) and represents the value that f(x) approaches as x gets closer to a (but not necessarily equal to a).
• If the limit exists and is equal to a finite value, the function is said to be continuous at that point.

Types of Limits:

• Left-hand Limit: lim(x→a-) f(x) represents the behavior of f(x) as x approaches a from the left side (values less than a).
• Right-hand Limit: lim(x→a+) f(x) represents the behavior of f(x) as x approaches a from the right side (values greater than a).
• Infinite Limits: If the limit of a function approaches positive or negative infinity as x approaches a certain value, it's called an infinite limit.

Continuity:

• A function is continuous at a point a if three conditions are met:
  1. The function is defined at a.
  2. The limit of the function as x approaches a exists.
  3. The limit of the function as x approaches a is equal to the value of the function at a.
• If a function is continuous at every point in its domain, it's called a continuous function.

Types of Discontinuity:

• Removable Discontinuity: A point where a function is not defined or has a hole, but it can be filled in to make the function continuous at that point.
• Jump Discontinuity: A point where the function has a sudden jump from one value to another.
• Infinite Discontinuity: A point where the function approaches positive or negative infinity.
• Asymptotic Discontinuity: A point where the function approaches a vertical asymptote.

Properties of Continuous Functions:

• Continuous functions satisfy several important properties, such as the intermediate value theorem, the extreme value theorem, and the composition of continuous functions is continuous.