This sub-organizer contains the following sections:
Prescribed Learning Outcomes
Suggested Instructional Strategies
Suggested Assessment Strategies
Recommended Learning Resources

PRESCRIBED LEARNING OUTCOMES

It is expected that students will understand the concept of a limit of a function, notation used, and be able to evaluate the limit of a function.

It is expected that students will:

• demonstrate an understanding of the concept of limit and notation used in expressing the limit of a function as x approaches
• evaluate the limit of a function
  • analytically
  • graphically
  • numerically
• distinguish between the limit of a function as x approaches a and the value of the function at x = a
• demonstrate an understanding of the concept of one-sided limits and evaluate one-sided limits
• determine limits that result in infinity (infinite limits)
• evaluate limits of functions as x approaches infinity (limits at infinity)
• determine vertical and horizontal asymptotes of a function, using limits
• determine whether a function is continuous at x = a

SUGGESTED INSTRUCTIONAL STRATEGIES

Students require a firm understanding of limits in order to fully appreciate the development of calculus.

• Although the concept of limit should be introduced at the beginning of the course, particular limits (e.g., ) need only be introduced as needed to deal with new derivatives.
• Using diagrams, introduce students in a general way to the two classic problems in calculus: the tangent line problem and the area problem and explain how the concept of limit is used in the analysis.
• Give students a simple function such as and have them: determine the slope of the tangent line to at the point (2,4) by evaluating the slope of the secant lines through (2,4) and (2.5, ƒ(2.5)), (2.1, ƒ(2.1)), (2.05, ƒ(2.05)), (2.01, ƒ(2.01)), then draw conclusions about the value of the slope of the tangent line
  • determine the area under above the x axis from by letting the number of rectangles be 4, 8, and 16, then inferring the exact area under the curve.
• Give students limit problems that call for analytic, graphical, and/or numerical evaluation, as in the following examples:
  • (evaluate analytically)
• (evaluate numerically, geometrically, and using technology)
• (evaluate analytically and numerically)
• (draw conclusions numerically)
• When introducing students to one-sided limits that result in infinity, have them draw conclusions about the vertical asymptote to , as in the following example: from , conclude that is the vertical asymptote for
• Have students explore the limits to infinity of a function and draw conclusions about the horizontal asymptote, as in the following example: because and because and are horizontal asymptotes for .

SUGGESTED ASSESSMENT STRATEGIES

Limits form the basis of how calculus can be used to solve previously unsolvable problems. Students should be able to demonstrate their knowledge of both the theory and application of limits in a variety of problem situations.

Observe

• While students are working on problems involving limits, look for evidence that they can:
  • distinguish between the limit of a function and the value of a function
  • Identify and evaluate one-sided limits
  • recognize and evaluate infinite limits and limits at infinity
  • determine any vertical or horizontal asymptote of a function
  • determine whether a function is continuous over a specified range or point
• To check students’ abilities to reason mathematically, have them describe orally the characteristics of limits in relation to one-sided limits, infinite limits, limits at infinity, asymptotes, and continuity of a function. To what extent do they give explanations that are mathematically correct, logical, and clearly presented.

Collect

• Assign a series of problems that require students to apply their knowledge of limits. Check their work for evidence that they:
  • clearly understood the requirements of the problem
  • used the appropriate method to evaluate the limit in the problem
  • verified that their solutions were accurate and reasonable

Self/Peer Assessment

• Have students explain concepts such as "limit" and "continuity" to each other in their own words.

RECOMMENDED LEARNING RESOURCES

Print Materials

• Calculus: Graphical, Numerical, Algebraic pp. 55-61, 65-77
• Calculus of a Single Variable Early Transcendental Functions, Second Edition
  pp. 63, 72, 75, 84, 96-97, 113, 227-228
• Single Variable Calculus Early Transcendentals, Fourth Edition
  Ch. 2 (Sections 2.2, 2.5, 2.6)

Multimedia

• Calculus: A New Horizon, Sixth Edition
  pp. 115, 118, 120, 122-124, 131, 148, 161
• Calculus of a Single Variable, Sixth Edition
  Ch. 1 (Section 1.2, 1.4, 1.5) Ch. 3 (Section 3.5)

© 2000 Copyright. All Rights Reserved. Curriculum Branch.
Maintained by: Mathematics Coordinator
Revised: December 5, 2000

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