Unit 1: Limits

Intro

  • The limit of a function is the main concept that distinguishes calculus from algebra and geometry (and all other math). Limits are fundamental to the study of calculus. Thus, it is beyond important to acquire a working understanding of limits before moving on to other topics in calculus such as derivatives and integrals.

  • In order to discover and create important ideas, definitions, formulas, and theorems in calculus, a firm understanding of limits is needed. Furthermore, limits can be used to understand the behavior of functions including concepts such as asymptotes (vertical and horizontal) and continuity.

Unit 1 Podcast

1.1 Determining Limits

    • Defining Limits

    • Limit Notation

    • Estimating Limits Graphically

    • Estimating Limits Numerically

    • Limit Properties

    • Direct Substitution

    • Dividing Out Technique

    • Rationalizing Technique

1.2 Limits and Graphing

    • Infinite Limits and Vertical Asymptotes

    • Limits at Infinity and Horizontal Asymptotes

    • Types of Discontinuities

    • Defining Continuity at a Point

    • Confirming Continuity over an Interval

    • Removing Discontinuities

    • Intermediate Value Theorem

Lesson Objectives

  • Represent limits analytically using correct notation.

  • Interpret limits expressed in analytic notation.

  • Estimate limits of functions.

  • Determine the limits of functions using limit theorems.

  • Determine the limits of functions using equivalent expressions for the function.

  • Interpret the behavior of functions using limits involving infinity.

  • The concept of a limit can be extended to include infinite limits.

  • Asymptotic and unbounded behavior of functions can be described and explained using limits.

  • The concept of a limit can be extended to include limits at infinity.

  • Limits at infinity describe end behavior.

  • Relative magnitudes of functions and their rates of change can be compared using limits.

  • Justify conclusions about continuity at a point using the definition.

  • Determine intervals over which a function is continuous.

  • Determine values of x or solve for parameters that make discontinuous functions continuous, if possible.

  • Explain the behavior of a function on an interval using the Intermediate Value Theorem.

  • Determine the limits of functions using the squeeze theorem.

Essential Knowledge

  • Common notation for limits

  • A limit can be expressed in multiple ways, including graphically, numerically, and analytically.

  • The concept of a limit includes one-sided limits.

  • Graphical information about a function can be used to estimate limits.

  • Because of issues of scale, graphically representations of functions may miss important function behavior.

  • A limit might not exist for some function at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.

  • Numerical information can be used to estimate limits.

  • Limits of sums, differences, products, quotients, and composite functions can be found using limit theorems.

  • It may be necessary or helpful to rearrange expressions into equivalent forms before evaluating limits.

  • The concept of a limit includes one-sided limits.

  • One-sided limits can be determined analytically and graphically.

  • A function is continuous at a point if the limit agrees on both sides of that point and is equal to the y-value at that point.

  • Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

  • A function is continuous on an interval if the function is continuous at each point in the interval.

  • Polynomial, rational, power, exponential, logarithmic, and trigonometric functions continuous on all points in their domains.

  • If the limit of a function exists at a discontinuity in its graph, then it is possible to remove the discontinuity by defining or redefining the value of the function at that point, so it equals the value of the limit of the function as approaches that point.

  • In order for a piecewise-defined function to be continuous at a boundary to the partition of its domain, the value of the expression defining the function on one side of the boundary must equal the value of the expressions defining the other side of the boundary, as well ass the value of the function at the boundary.

  • If there is a continuous function on a closed interval, then the Intermediate Value Theorem guarantees that there is at least one x-value that has a y-value between the endpoint y-values on the interval.

  • The limit of a function may be found by using the squeeze theorem.