01 Limits

The first "big idea" in calculus is the limit of a function. Incremental changes in a natural system are measured over small intervals. As the interval becomes smaller, our accuracy in modeling the system improves. The mathematical concept of a limit allows us to make the interval as small as we choose - essentially infinitesimal. The limit provides the mathematical foundation for the other two big ideas of calculus (the derivative and the integral).

You will be assessed on the following standards.

1.5 Can estimate limits from graphs and tables

1.6 Can compute limits using the limit laws and algebra

Learning Goals for Chapter 1 - Limits

Videos and Suggested Homework (HW)

1.4 The Tangent and Velocity Problem

  • The tangent line viewed as the limit of secant lines.

  • The concepts of average versus instantaneous velocity, described numerically, graphically, and in physical terms.

  • The tangent line as the line obtained by “zooming in” on a smooth function; local linearity.

  • Approximating the slope of the tangent line using slopes of secant lines.

We will analyze and discuss what information the tangent to a curve gives us about a function.

Introduction to Limits

HW: Finding Limits Numerically

1.5 The Limit of a Function

  • Estimating limits from graphs and tables.

  • Understanding asymptotes and describing asymptotic behavior in terms of limits involving infinity.

  • Using computing devices and limits.

We will use calculators and Excel to help us understand the concept of a limit and to estimate limiting values for functions.

Estimating Limits from Graphs

HW: One-Sided Limits from Graphs

HW: Two-sided Limits from Graphs

1.6 The Limit Laws

  • The algebraic computation of limits: manipulating algebraically, examining left- and right-hand limits, using the limit laws to break monstrous functions into pieces, and analyzing the pieces.

  • The evaluation of limits from graphical representations.

  • Examples where limits don’t exist (using algebraic and graphical approaches).

  • The computation of limits when the limit laws do not apply, and the use of direct substitution property when they do.

We will learn how to use the limit laws to determine exact values of limits analytically using algebra.

Finding Limits Algebraically

HW: Two-sided Limits using Algebra

HW: Two-sided Limits Using Advanced Algebra

1.8 Continuity

  • The graphical and mathematical definitions of continuity, and the basic principles.

  • Examples of discontinuity.

  • The Intermediate Value Theorem: mathematical statement, graphical examples, and applied examples.

You will understand the concept of continuity, and its formal mathematical definition.

Continuity Using Limits

HW: Continuity

Showing 2 items
Minimum Suggested Practice ProblemsComments
Minimum Suggested Practice ProblemsComments
[u1.4: 1, 3, 5, 7] and [u1.5: 1, 5, 7, 15, 29]  
[u1.6: 1, 3, 5, 11-21 odd, 63] and [u1.8: 3, 11, 15, 51]  
Showing 2 items
Nikhil Joshi,
Nov 14, 2014, 12:36 PM