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  • Modern Books
    • A Quick Review of Pre-Calculus
    • Single Variable Calculus
    • Multivariable Calculus
    • Numerical Methods
    • Mechanics of Materials
  • Classic Books
    • Calculus Made Easy
    • A Course of Pure Mathematics
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Indeterminate Forms and L’Hôpital’s Rule

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  3. 6. Applications of Differentiation>
  4. Indeterminate Forms and L’Hôpital’s Rule

Calculus

  • 1. Review of Fundamentals
    • 1.1 What Is Algebra?
    • 1.2 Sets
    • 1.3 Sets of Numbers
    • 1.4 Inequalities
    • 1.5 Absolute Value
    • 1.6 Intervals
    • 1.7 Laws of Exponents
    • 1.8 The Logarithm
    • 1.9 Equations and Identities
    • 1.10 Polynomials
    • 1.11 Dividing Polynomials
    • 1.12 Special Product Formulas
    • 1.13 Factorization
    • 1.14 Fractions
    • 1.15 Rationalizing Binomial Denominators
    • 1.16 Coordinates in a Plane
    • 1.17 Graphs of Equations
    • 1.18 Straight Lines
    • 1.19 Solutions and Roots
    • 1.20 Other Types of Equations
    • 1.21 Solving Inequalities
    • 1.22 Absolute Value Equations and Inequalities
    • 1.23 Sigma Notation
  • 2. Functions
    • 2.1 Constants, Variables, and Parameters
    • 2.2 The Concept of a Function
    • 2.3 Natural Domain and Range of a Function
    • 2.4 Graphs of Functions
    • 2.5 Vertical Line Test
    • 2.6 Domain and Range Using Graph
    • 2.7 Piecewise-Defined Functions
    • 2.8 Equal Functions
    • 2.9 Even and Odd Functions
    • 2.10 Examples of Elementary Functions
    • 2.11 Transformations of Functions
    • 2.12 Algebraic Combination of Functions
    • 2.13 Composition of Functions
    • 2.14 Increasing or Decreasing Functions
    • 2.15 One-to-One Functions
    • 2.16 Inverse Functions
  • 3. Transcendental functions
    • 3.1 Exponential Functions
    • 3.2 Logarithmic Functions
    • 3.3 Trigonometric Functions
      • 3.3.1 Angles
      • 3.3.2 Basics of Trigonometric Functions
      • 3.3.3 Trigonometric Identities
      • 3.3.4 Periodicity and Graphs of Trigonometric Functions
    • 3.4 Inverse Trigonometric Functions
  • 4. Limits and Continuity
    • 4.1 Concept of a Limit
    • 4.2 The Precise Definition of a Limit
    • 4.3 One-Sided Limits
    • 4.4 Theorems for Calculating Limits
    • 4.5 The indeterminate Form 0/0
    • 4.6 Infinite Limits
    • 4.7 Limits at Infinity
    • 4.8 Continuity
    • 4.9 Properties of continuous functions
    • 4.10 The Other Indeterminate Forms
    • 4.11 Asymptotes
    • 4.12 The number e
  • 5. Differentiation
    • 5.1 The Derivative Concept
    • 5.2 Geometric Interpretation of the Derivative as a Slope
    • 5.3 Graphing the Derivative
    • 5.4 Differentiability and Continuity of Functions
    • 5.5 One-sided Derivatives
    • 5.6 When a Function Is Not Differentiable at a Point
    • 5.7 Differentiation Rules
    • 5.8 Derivatives of The Trigonometric Functions
    • 5.9 Higher Derivatives
    • 5.10 More About the Leibniz Notation for Higher Derivatives
    • 5.11 Implicit Differentiation
    • 5.12 Derivatives of Inverse Functions
    • 5.13 Derivatives of the Inverse Trigonometric Functions
    • 5.14 Derivatives of Logarithmic Functions
    • 5.15 Derivatives of Exponential Functions
    • 5.16 Hyperbolic Functions and Their Derivatives
  • 6. Applications of Differentiation
    • 6.3 Related rates
    • 6.4 Extreme values of functions
    • 6.5 Rolle’s theorem
    • 6.6 The Mean-Value Theorem for Derivatives
    • 6.7 Increasing and decreasing functions
    • 6.8 Concavity and Points of Inflection
    • 6.9 First Derivative Test for Local Extrema
    • 6.10 Second Derivative Test for Local Extrema
    • 6.11 L’Hôpital’s Rule for Indeterminate Limits
    • 6.12 Curve Sketching
    • 6.13 Optimization
  • 7. Indefinite Integrals
    • 7.1 Definition
    • 7.2 Rules for Integrating Standard Elementary Forms
    • 7.3 Constant of integration
    • 7.4 Integration by Substitution
  • 8. Definite Integrals
    • 8.1 Definition of the Definite Integral
    • 8.2 Properties of definite integrals
    • 8.3 The Fundamental Theorem of Calculus
    • 8.4 Derivatives of Integrals
  • 9. Applications of Integration
    • 9.1 The Area Between Two Curves
    • 9.2 Volumes of Solids of Revolution: the Disk and Washer Methods
    • 9.3 Volumes of Solids with Known Cross-Sections: The Slice Method
    • 9.4 Volumes: The Shell Method
    • 9.5 Arc Length
    • 9.6 Work
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