Education Technology

Calculus: A Tale of Two Lines

by Texas Instruments


  • Determine limits of ratios of functions appearing linear using approximation
  • Recognize the relationship between the ratio of slopes of linear functions and the ratios of the values of linear functions
  • Apply the preceding ideas to non-linear functions by recognizing the relationships between local linearity, slopes of functions, and the derivatives of functions
  • Learn and apply l’Hôpital’s Rule


  • limit
  • derivative
  • differentiable

About the Lesson

This lesson involves demonstrating a visual justification for l’Hôpital’s Rule as applied to 0/0 forms. As a result, students will:

  • Begin with a zoomed-in graph of two functions, displaying both functions as linear. They will observe that the ratio of the slopes of the functions is the same as the ratio of the y-values of the function near the point where both are 0.
  • Zoom out on the functions, revealing two non-linear functions. They will note that the limit of the quotients of the functions at their point of intersection cannot be determined algebraically.
  • Recognize that the slope of the zoomed-in functions is the same as the derivative of the functions at that point, and use that information to justify l’Hôpital’s Rule.