Limits and Continuity

Introduction to Limits

Limits are fundamental to calculus, describing the behavior of functions as inputs approach specific values. Understanding limits is essential for defining derivatives and integrals.

• Definition: The limit of a function f(x) as x approaches a value a is the value that f(x) gets closer to as x gets closer to a.

• Notation:

• Key Properties:

  • Limits can be evaluated by direct substitution, factoring, or rationalizing.

  • Indeterminate forms (e.g., ) require algebraic manipulation.

• Example:

• Laws of Limits:

  • (if denominator ≠ 0)

• Special Limits:

Continuity

A function is continuous at a point if its limit exists and equals the function value at that point.

• Definition: f(x) is continuous at x = a if

• Types of Discontinuity: Removable, jump, and infinite discontinuities.

Techniques of Differentiation

Differentiation from First Principles

Differentiation measures the rate at which a function changes. The derivative from first principles uses the definition of the limit.

• Definition: The derivative of f(x) at x is

• Example: For ,

Rules of Differentiation

Several rules simplify the process of finding derivatives for various types of functions.

• Sum Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

• Example:

Differentiating Trigonometric, Exponential, and Logarithmic Functions

Special rules apply to these functions due to their unique properties.

• Trigonometric Functions:

• Exponential Functions:

• Logarithmic Functions:

• Example:

Applications of Differentiation

Finding Maxima, Minima, and Points of Inflection

Derivatives are used to locate critical points where functions reach local maximum or minimum values, and points of inflection where the curvature changes.

• Critical Points: Occur where or is undefined.

• Second Derivative Test: If , minimum; if , maximum.

• Point of Inflection: Where and changes sign.

• Example: For , critical points at and .

Rates of Change

Derivatives represent rates of change in various contexts, such as velocity, acceleration, and growth rates.

• Velocity: , where s is displacement.

• Acceleration:

• Example: If , then and .

Summary Table: Differentiation Rules

Implicit Differentiation

Introduction

Implicit differentiation is used when functions are defined implicitly rather than explicitly.

• Method: Differentiate both sides of the equation with respect to x, treating y as a function of x.

• Example: For ,

Additional info:

• These notes cover core calculus topics relevant for college-level study, including limits, differentiation techniques, and applications.

• Examples and tables are expanded for clarity and completeness.