Calculus: A Tale of Two Lines

Objectives

• 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

Vocabulary

• 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.