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    • ▾Calculus 1
      • ▸Limits and continuity
        • •Intro to calc 1
        • •Estimating limits from graphs
        • •Estimating limits from tables
        • •Intro to continuity
        • •Finding and classifying discontinuities
        • •Finding the value that makes the function continuous
      • ▸Derivatives: definition and basic derivative rules
        • •Proving differentiable implies continuous
        • •Derivative of a function from first principles
        • •Finding the tangent or normal line through the given point
        • •Constant rule
        • •Constant multiple rule for derivatives
        • •Sum, difference, and power rules for derivatives
        • •Product rule
        • •Proving the product rule
        • •Quotient rule
        • •Derivatives of trig functions
        • •Proving properties of even and odd functions (calculus 1)
        • •Leibniz's derivative notation
        • •Logarithmic functions
      • ▸Derivatives: composite, implicit, and inverse functions
        • •Derivative at a point from first principles
        • •Derivatives of exponential functions
        • •Derivatives of logarithmic functions
        • ▸Derivatives of hyperbolic and inverse hyperbolic functions
          • •Using the derivatives of the hyperbolic functions
          • •Proving the derivatives of the hyperbolic functions
          • •Proving the derivatives of the inverse hyperbolic functions
        • •Derivatives of inverse trig functions
        • •Derivatives of transcendental functions
        • •Derivatives of trigonometric functions under transformations
        • •Derivative using u-substitution
        • •Determining derivatives from graphs
        • •Higher order derivatives
        • •Chain rule
        • •Derivatives of inverse functions
        • •Choosing the derivative rules to use
      • ▾Applying derivatives to analyze functions
        • •Applications of trigonometric derivatives
        • •Extreme value theorem
        • •First derivative test
        • •Mean value theorem
        • ▸Monotonic intervals and functions
          • •Intro
          • •Proofs
        • •Related rates
        • •Second derivative test
        • •Concavity
        • •Factoring a cubic polynomial with a double root
        • •Graphing using derivatives
        • ▸Special points
          • ▸Finding critical values from graphs
            • •Lesson
            • •Practice
          • •Finding critical values using derivatives
          • ▸Inflection points
            • •Graphically finding points of inflection
            • •Using the second derivative to find points of inflection
          • •Classifying stationary points
          • •Sketching polynomials using stationary points
        • •Bisection method
        • •Newton's method
        • •False position method
        • •L'Hopital's rule
      • ▸Integrals
        • •Basic integration problems
        • ▸Integration by u-substitution
          • ▸Integration by u-substitution (less difficult)
            • •Lesson
            • •Practice
          • •Integration by u-substitution (more difficult)
          • •Integrating exponential functions by u-substitution
        • •Visually determining antiderivative
      • ▸Applications of integrals
        • •Proving the formula for the area of a circle
        • •Area between two curves
        • ▸Solids of revolution
          • •Disc and washer methods (circular cross sections)
          • •Volumes of solids with known cross sections
          • •Shell method
          • •Volume of a sphere (calculus \(2\))
          • •Volume of a cone (calculus 2)
     › Calculus 1 › Applying derivatives to analyze functions

    L'Hopital's rule

    In this section, you will learn what L'Hopital's rule is, and how to use it. You will see that some problems require using L'Hopital's rule more than once. Before attempting this section, you should be excellent at finding derivatives.

    Typeset step-by-step solutions.
    \(\displaystyle \lim_{x \to 0}\left(\dfrac{e^x - 1}{x}\right)\)
    \(1\)
    72370Find the Limit Using L'Hopitals Rule | Calculus #Shorts
    Prof. Redden
    \(\displaystyle \lim_{x \to 0}\left(\dfrac{e^x - 1}{x^3}\right)\)
    \(\infty\)
    21327Calc Unit 8 Apply L'Hopital's Rule to evaluate the Limit
    Brian McLogan
    \(\displaystyle \lim_{x \to 0}\left(\dfrac{2x^2}{\cos x - 1}\right)\)
    \(-4\)
    72371Find the Limit using L'Hopitals Rule TWICE Calculus #Shorts
    Prof. Redden
    \(\displaystyle \lim_{x \to 1}\left(\dfrac{x^9 - 1}{x^5 - 1}\right)\)
    \(\dfrac{9}{5}\)
    28277l'hospital Example 1 zero/zero
    Mathbyfives
    \(\displaystyle\lim_{x \to \infty}\dfrac{x^2}{e^x}\)
    \(0\)
    3271L'hopital's rule
    The Organic Chemistry Tutor
    \(\displaystyle\lim_{x \to \infty}\dfrac{\ln x}{x}\)
    \(0\)
    3271L'hopital's rule
    The Organic Chemistry Tutor
    \(\displaystyle\lim_{x \to 0}\dfrac{\sin(7x)}{\sin(4x)}\)
    \(\dfrac{7}{4}\)
    3271L'hopital's rule
    The Organic Chemistry Tutor
    \(\displaystyle\lim_{x \to 0}\dfrac{\sin(8x)}{3x}\)
    \(\dfrac{8}{3}\)
    3271L'hopital's rule
    The Organic Chemistry Tutor
    \(\displaystyle\lim_{x \to \infty}x\ln x\)
    \(\infty\)
    3271L'hopital's rule
    The Organic Chemistry Tutor
    \(\displaystyle\lim_{x \to \infty}x^{1 / x}\)
    \(1\)
    3271L'hopital's rule
    The Organic Chemistry Tutor
    \(\displaystyle\lim_{x \to 0}(1 - 2x)^{1 / x}\)
    \(e^{-2}\)
    3271L'hopital's rule
    The Organic Chemistry Tutor
    \(\displaystyle\lim_{x \to 0} \dfrac{\sin x}{x}\)
    \(1\)
    19442L'Hopital's Rule sine x over x
    Brian McLogan

    Conclude by giving your students these challenges:

    • Jugs of Wine by NRICH
    • What Does Random Look Like? by NRICH
    • Point of Contact Triangle

    Cubist Cuts by NRICH: The first part of this challenge also appears in Mathematics, A Human Endeavor by Harold R. Jacobs, on page 31. One of their diagrams is helpful, and doesn't appear on the NRICH page.

    A farmer has 36 pieces of fence. He wants to arrange them into a rectangle or a regular polygon. Which shapes can he make? This problem is based on Jo Boaler's 36 Fences problem, which can be found on page 51 of Jo Boaler's Studies in Mathematical Thinking.

    YouTube videos

    • 1398L'Hopital's Rule Lesson 8 Examples Calculus 2 BC (includes small correction)
      ProfRobBob
    • 28277l'hospital Example 1 zero/zero
      Mathbyfives
    • 14035How to apply L'Hopital's Rule to evaluate the limit
      Brian McLogan
    • 44271Finding a Limit Using L'Hopital's Rule (e^x - 1)/x as x approaches zero
      The Math Sorcerer