Limits Continuity and Differentiability

Limits:

Definition: A limit represents the value that a function approaches as the input approaches a certain value.

Notation: The limit of a function f(x) as x approaches a is denoted by limxaf(x).

Properties:

  • Limit Laws:
    • Sum/Difference Law: limxa[f(x)±g(x)]=limxaf(x)±limxag(x)
    • Product Law: limxa[f(x)g(x)]=limxaf(x)limxag(x)
    • Quotient Law: limxaf(x)g(x)=limxaf(x)limxag(x)
    • Power Law: limxa[f(x)]n=[limxaf(x)]n (where n is a constant)

Types of Limits:

  • One-sided limits: Limits calculated from one direction (left or right) of a specific point.
  • Infinite limits: When the limit of a function approaches infinity or negative infinity.

Continuity:

Definition: A function is continuous at a point if the function is defined at that point, the limit of the function exists at that point, and the limit equals the value of the function at that point.

Types of Discontinuities:

  • Removable Discontinuity: A hole in the graph that can be filled by redefining the function at that point.
  • Jump Discontinuity: The function "jumps" from one value to another at a specific point.
  • Infinite Discontinuity: Asymptotic behavior where the function approaches infinity or negative infinity at a certain point.

Continuity Theorems:

  • Intermediate Value Theorem: If a function f(x) is continuous on the interval [a,b] and f(a) and f(b) have different signs, then there exists at least one value c in the interval [a,b] such that f(c)=0.
  • Extreme Value Theorem: If a function f(x) is continuous on a closed interval [a,b], then it has both a maximum and minimum value on that interval.

Differentiability:

Definition: A function is differentiable at a point if the derivative at that point exists.

Differentiability Conditions:

  • Differentiable implies Continuous: If a function is differentiable at a point, it must be continuous at that point.
  • Differentiation Rules:
    • Power Rule: ddx[xn]=nxn1
    • Sum/Difference Rule: ddx[f(x)±g(x)]=ddx[f(x)]±ddx[g(x)]
    • Product Rule: ddx[f(x)g(x)]=f(x)g(x)+f(x)g(x)
    • Quotient Rule: ddx[f(x)g(x)]=f(x)g(x)f(x)g(x)[g(x)]2

Differentiability Theorems:

  • Rolle's Theorem: If a function f(x) is continuous on the closed interval [a,b], differentiable on the open interval (a,b), and f(a)=f(b), then there exists at least one c in the interval (a,b)such that f(c)=0.
  • Mean Value Theorem: If a function f(x) is continuous on the closed interval [a,b], differentiable on the open interval (a,b), then there exists at least one c in the interval (a,b) such that f(b)f(a)ba=f(c).