Limits Involving Trigonometric Functions

Introduction

Limits involving trigonometric functions are fundamental in calculus, especially when analyzing the behavior of functions near specific points. These limits often appear in the computation of derivatives and in solving real-world problems involving periodic phenomena.

Key Trigonometric Limits

• Limit of sin(x)/x as x approaches 0:

• Limit of (cos(x) - 1)/x as x approaches 0:

• Limit of sin(ax)/x as x approaches 0:

• Limit of sin(x)/x for higher multiples:

• Limit of sin(kx)/kx as x approaches 0:

Examples

• Example 1: Since , by substitution.

• Example 2: The limit is $2$.

• Example 3: The limit is .

Derivatives of Trigonometric Functions

Introduction

The derivatives of trigonometric functions are essential tools in calculus. They are used to analyze rates of change in periodic phenomena and are foundational for solving many calculus problems.

Basic Derivatives

• Derivative of sin(x):

• Derivative of cos(x):

• Derivative of tan(x):

• Derivative of cot(x):

• Derivative of sec(x):

• Derivative of csc(x):

Summary Table: Derivatives of Trigonometric Functions

Derivatives Using the Limit Definition

• Limit definition of derivative:

• Example for sin(x): Using the identity :

  • Substitute and simplify:

  • As , and

  • Result:

Trigonometric Addition Formulas

• Sine Addition Formula:

• Cosine Addition Formula:

Quotient Rule for Derivatives

The quotient rule is used to differentiate functions of the form .

• Quotient Rule:

• Example: Differentiate

  • ,

  • ,

  • Apply quotient rule:

Summary of Derivative Rules

• Product Rule:

• Quotient Rule:

Applications and Additional Examples

Applications

• Trigonometric derivatives are used in physics to model oscillatory motion, such as waves and pendulums.

• Limits involving trigonometric functions are essential in evaluating indeterminate forms and in the study of continuity and differentiability.

Additional info:

• Some formulas and steps were inferred from context and standard calculus knowledge, as the original notes were fragmented.

• All key trigonometric limits and derivative rules are included for completeness.