Calculus Basics
  • Introduction
  • ▶️Limit & Continuity
    • Limit properties & Limits of Combined Functions
    • Limits at infinity
    • All types of discontinuities
  • ▶️Differential Calculus
    • Differentiability
    • Local linearity & Linear approximation
    • Basic Differential Rules
    • Chain Rule
    • Derivatives of Trig functions
    • Implicit differentiation
    • Higher Order Derivatives
    • Derivative of Inverse functions
    • Derivative of exponential functions
    • Existence Theorems
    • L'Hopital's Rule
    • Critical points
      • Extrema: Maxima & Minima
      • Concavity
      • Inflection Point
    • Second Derivative Test
    • Anti-derivative
    • Analyze Function Behaviors with Derivatives
    • Optimization
    • Applications of Derivatives
      • Motion problems
      • Planar motion
  • ▶️Integral Calculus
    • Definite Integrals
    • Antiderivatives
    • Fundamental Theorem of Calculus (FTC)
    • Basic Integral Rules
    • Calculate Integrals
    • Integration using Trig identities
    • Improper Integral
    • U-substitution → Chain Rule
    • Integrate by Parts → Product Rule
    • Partial fractions → Log Rule
    • Trig-substitutions → Trig Rule
    • Average Value of Functions
  • ▶️Differential Equations
    • Parametric Equations Differentiation
    • Separable Differential Equations
    • Specific antiderivatives
    • Polar Curve Functions (Differential Calc))
    • Logistic Growth Model
    • Slope Field
    • Euler's Method
  • ▶️Applications of definite integrals
  • ▶️Series (Calculus)
    • Infinite Seires
    • Infinite Geometric Series
    • Convergence Tests
      • nth Term Test
      • Integral Test
      • p-series Test
      • Comparison Test
      • Ratio Test
      • Root Test
      • Alternating Series Test
    • Absolute vs. Conditional Convergence
      • Error Estimation of Alternating Series
      • Error Estimation Theorem
      • Interval of Convergence
    • Power Series
      • Taylor Series
      • Maclaurin Series
      • Lagrange Error Bound
      • Finding Taylor series for a function
      • Function as a Geometric Series
      • Maclaurin Series of Common functions
      • Euler's Formula & Euler's Identity
  • Multivariable functions
    • Parametric Functions
    • Partial derivatives
    • Gradient
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  • Example of 0/0
  • Example of ∞/∞
  • Example of 1^∞
  • L'Hopital Rule for Composite functions
  • Example of composite exponential function
  1. Differential Calculus

L'Hopital's Rule

PreviousExistence TheoremsNextCritical points

Last updated 6 years ago

LHopital's Rule helps us to find the limit of an Undefined limits, like 0/0, ∞/∞ and such. It's quite simple to apply and very convenient to solve some problems.

From my experience, the L'Hopital's Rule is so often been used that we didn't even realize. Actually it's been used almost every time when we are to evaluate the LIMITS OF RATIONAL EXPRESSIONS.

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Example of 0/0

Example of ∞/∞

Example of 1^∞

L'Hopital Rule for Composite functions

Example of composite exponential function

  • Direct plug in the x=2 and get limit = 1^∞, which is an indeterminate form

  • So we're gonna apply the Log Power Rule to take down the power:

  • We use Natural Log for doing this:

  • And now we can apply the L'Hopital Rule for ln(y):

  • From ln(y) we could get limit of y:

Find the limit: Solve:

Find the limit: Solve:

Solve:

▶️
Refer to L'Hôpital's rule
▶ Back previous note on: Asymptote of Rational Expressions
▶ Practice at Khan academy: Disguised derivatives
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