Limits and Limit Theorems in Calculus

Definition of a Limit

• A limit is a unique real value that a function approaches as the variable x approaches a constant a.
• The limit, denoted by L, is the value that the function f(x) will approach as x approaches a.

Limit Theorems

• The limit of a function f(x) as x approaches a is denoted by lim f(x) = L.
• The limit exists if we find a single real function value upon applying the limit theorems.
• If lim f(x) = L1 and lim f(x) = L2, then L1 = L2.

One-Sided Limits

• The limit of f(x) as x approaches 2 from the right is 4.
• The limit of f(x) as x approaches 2 from the left is 4.

Informal Definition of a Limit

• The limit of a function f of a single variable x is the unique real value that f(x) approaches as x approaches a constant a.
• The limit is denoted by L, and is written as lim f(x) = L.

Limit Theorems (Examples)

• Example 1: lim (2x - 1) / (x - 1) = 2
• Example 2: lim (x^2 + x - 2) / (2x - 1) = 4
• Example 3: lim (x^2 + x - 2) / (x - 2) = 4

Existence of Limits

• The limit exists if we find a single real function value upon applying the limit theorems.
• If lim f(x) = L1 and lim f(x) = L2, then L1 = L2.

Uniqueness of Limits

• If the limit exists, it is unique.
• If lim f(x) = L1 and lim f(x) = L2, then L1 = L2.

Rational Functions

• A rational function is a function of the form f(x) = h(x) / g(x), where h(x) and g(x) are polynomial functions.
• The limit of a rational function as x approaches a is denoted by lim f(x) = L.

Radical Functions

• A radical function is a function of the form f(x) = x^n, where n is a real number.
• The limit of a radical function as x approaches a is denoted by lim f(x) = L.

Examples of Limit Theorems

• Example 1: lim x^2 = 4
• Example 2: lim x^3 = 8
• Example 3: lim x^4 = 16

Infinite Limits

• The limit of a function as x approaches a is infinite if the function values increase or decrease without bound as x approaches a.
• The limit of a function as x approaches a is denoted by lim f(x) = ∞ or lim f(x) = -∞.