Calculus - Introduction to Limits

Calculus - Limits

Introduction to Limits

• A limit represents the behavior of a function as the input (or x-value) approaches a specific point.
• It is denoted by the symbol lim and is read as "the limit as x approaches a of f(x)".
• The concept of limits is central to calculus and is used to define derivatives and integrals.

Notation

• lim x→a f(x) = L means that as x approaches a, the value of f(x) approaches L.
• x→a means x approaches a, but does not necessarily equal a.
• f(x) → L means the value of f(x) approaches L.

Properties of Limits

• Linearity: The limit of a sum is the sum of the limits: lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
• Homogeneity: The limit of a constant multiple is the constant multiple of the limit: lim x→a [k \* f(x)] = k \* lim x→a f(x)
• Sum of Limits: The limit of a sum is the sum of the limits: lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
• Product of Limits: The limit of a product is the product of the limits: lim x→a [f(x) \* g(x)] = lim x→a f(x) \* lim x→a g(x)
• Chain Rule: The limit of a composite function is the composite of the limits: lim x→a f(g(x)) = f(lim x→a g(x))

Types of Limits

• One-Sided Limits: A limit that approaches a from one side, denoted by x→a+ or x→a-.
• Two-Sided Limits: A limit that approaches a from both sides, denoted by x→a.
• Infinite Limits: A limit that approaches infinity or negative infinity, denoted by lim x→a f(x) = ∞ or lim x→a f(x) = -∞.

Important Limit Theorems

• Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) and lim x→a f(x) = lim x→a h(x) = L, then lim x→a g(x) = L.
• Sandwich Theorem: If f(x) ≤ g(x) ≤ h(x) and lim x→a f(x) = L and lim x→a h(x) = L, then lim x→a g(x) = L.

Introduction to Limits

• A limit describes a function's behavior as the input approaches a specific value, denoted by lim.
• Essential for defining derivatives and integrals in calculus.

Notation

• lim x→a f(x) = L: Indicates f(x) approaches L as x approaches a.
• x→a: x may approach but does not need to equal a.
• f(x) → L: The function f(x) approaches the value L.

Properties of Limits

• Linearity: lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x).
• Homogeneity: lim x→a [k * f(x)] = k * lim x→a f(x).
• Product of Limits: lim x→a [f(x) * g(x)] = lim x→a f(x) * lim x→a g(x).
• Chain Rule: For composite functions, lim x→a f(g(x)) = f(lim x→a g(x)).

Types of Limits

• One-Sided Limits: Approaches from one side, represented as x→a+ or x→a-.
• Two-Sided Limits: Approaches from both sides, denoted by x→a.
• Infinite Limits: Approaches infinity or negative infinity; written as lim x→a f(x) = ∞ or lim x→a f(x) = -∞.

Important Limit Theorems

• Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) and both f(x) and h(x) approach L, then g(x) also approaches L.
• Sandwich Theorem: Similar to the Squeeze Theorem, confirming that if both bounding functions converge to L, so does g(x).

This quiz covers the fundamental concept of limits in calculus, focusing on how to interpret the behavior of functions as they approach specific values. Learn the essential notation and principles that form the foundation for more advanced calculus topics such as derivatives and integrals.