Limits

Definition of Limit:

Definition: The limit of a function describes the behavior of the function as its input values approach a certain point. It is a fundamental concept in calculus used to define continuity, derivatives, and integrals.

Symbolic Representation: Mathematically, the limit of a function f(x) as x approaches a value c is denoted as: limxcf(x)=L

Understanding the Notation:

  • xc signifies that the input variable x is getting arbitrarily close to c.
  • L represents the limit value. If the limit exists, L is the value that f(x) approaches as x gets closer to c.

Informal Definition: The limit of a function f(x) at x=c exists if, as x gets arbitrarily close to c, the values of f(x) get arbitrarily close to a single value L.

Formal Definition (ε-δ Definition): For all ϵ>0, there exists a δ>0 such that if 0<xc<δ, then f(x)L<ϵ.

  • ϵ and δ are small positive numbers.
  • ϵ represents the closeness of f(x) to the limit L.
  • δ represents the closeness of x to c.

Properties of Limits:

  1. Limit Laws: The basic properties governing limits include addition, subtraction, multiplication, division, and composition of functions.
  2. Existence of Limits: A limit exists if and only if the left-hand limit and right-hand limit at that point are equal.
  3. Infinite Limits: A limit can also tend towards positive or negative infinity if the function grows without bound.

Left-Hand and Right-Hand Limits:

Definition:

  • Left-Hand Limit: The left-hand limit of a function f(x) as x approaches a value c from the left side (or from values smaller than c) is denoted as limxcf(x).

  • Right-Hand Limit: The right-hand limit of a function f(x) as x approaches a value c from the right side (or from values greater than c) is denoted as limxc+f(x).

Left-Hand Limit (limxcf(x)):

  • Represents the behavior of the function as x approaches c from the left side.
  • x values are strictly less than c.
  • Symbolically, if the left-hand limit exists and equals L, it implies that as x gets arbitrarily close to c from the left, f(x) approaches L.

Right-Hand Limit (limxc+f(x)):

  • Represents the behavior of the function as x approaches c from the right side.
  • x values are strictly greater than c.
  • Symbolically, if the right-hand limit exists and equals M, it implies that as x gets arbitrarily close to c from the right, f(x) approaches M.

Existence of Overall Limit:

  • For the overall limit limxcf(x) to exist, both the left-hand and right-hand limits must exist and be equal. Mathematically, limxcf(x)=limxc+f(x)=L.

Useful Insights:

  • Jump Discontinuity: If the left-hand and right-hand limits exist but are not equal (i.e., limxcf(x)limxc+f(x), it signifies a jump discontinuity in the function at x=c.

  • Infinite Limits: One-sided limits can also approach positive or negative infinity. For instance, limxcf(x)= indicates that as x approaches c from the left, f(x) grows unbounded.

Algebra of Limits:

Overview: The algebra of limits involves applying basic arithmetic operations to functions that are subject to limits. It allows the manipulation of functions in limit calculations using properties similar to those of regular algebra.

Key Properties:

  1. Limit of a Sum:

    • limxc[f(x)+g(x)]=limxcf(x)+limxcg(x)
    • The limit of the sum of two functions is the sum of their individual limits, provided the limits of the functions exist.
  2. Limit of a Difference:

    • limxc[f(x)g(x)]=limxcf(x)limxcg(x)
    • The limit of the difference of two functions is the difference of their individual limits, given that the limits of the functions exist.
  3. Limit of a Product:

    • limxc[f(x)g(x)]=limxcf(x)limxcg(x)
    • The limit of the product of two functions is the product of their individual limits, assuming the limits of the functions exist.
  4. Limit of a Quotient:

    • limxc[f(x)g(x)]=limxcf(x)limxcg(x)
    • The limit of the quotient of two functions is the quotient of their individual limits, provided the limits of both functions exist and the denominator function does not approach zero.
  5. Limit of a Constant Multiple:

    • limxc[kf(x)]=klimxcf(x)
    • A constant multiple can be factored out of the limit operation.