Unit 1 of 10

Limits

  • 13 lessons
  • Parts 1 to 26

A limit describes what a function's values approach as the input gets close to some point, even when the function is not defined there. Almost everything else in calculus, from derivatives to integrals to series, is a limit in disguise.

This unit starts with the questions that made limits necessary, velocity at an instant and the slope of a curve, then builds the idea from intuition up to the formal epsilon and delta definition, the limit laws, and limits that involve infinity.

Every lesson is a pair of videos. The theory part explains the idea and the practice part works through problems. Each lesson page has the definitions and an example for every key term.

Motivation and Limits

  1. Velocity and Tangent

    How the speed of a moving object at one instant and the slope of a curve at one point lead to the same question, and why answering it needs limits.

    Theory part 1Practice part 2

  2. Intuitive Definition of Limit

    What it means for a function to approach a number as its input approaches a point, how to read limits from tables and graphs, and the ways a limit can fail to exist.

    Theory part 3Practice part 4

Limits of Functions

  1. One-Sided Limits

    How to describe what a function approaches from the left or from the right of a point, and how to read one-sided limits of piecewise functions and at the edge of a domain.

    Theory part 5Practice part 6

  2. Two-Sided Limits

    When a function has a single limit at a point, why it can only have one, how it relates to the two one-sided limits, and what changes when the function is defined on one side only.

    Theory part 7Practice part 8

Formal Definitions

  1. The Formal Definition of a Limit

    The epsilon-delta definition of a limit, why the order of its quantifiers matters, how to choose delta in a proof, and how to show that a number is not the limit.

    Theory part 9Practice part 10

  2. The Formal Definition of One-Sided Limits

    The epsilon-delta definitions of left-hand and right-hand limits, how they handle the endpoints of a domain, and how to prove a one-sided limit or show that it fails.

    Theory part 11Practice part 12

Limits of Functions

  1. Limit Laws

    The rules that compute limits of sums, differences, products, quotients, powers and roots from simpler limits, the conditions each rule needs, and how one of them is proved.

    Theory part 13Practice part 14

  2. Limit Theorems

    Direct substitution for polynomials and rational functions, the squeeze theorem, how limits respect inequalities, and the algebra that rescues limits of the form zero over zero.

    Theory part 15Practice part 16

  3. Special Limits

    Why sin x over x tends to 1 when x is in radians, the related limit for one minus cosine, the basic exponential and logarithm limits at zero, and how to use them without circular reasoning.

    Theory part 17Practice part 18

Limits at Infinity and Infinite Limits

  1. Limits at Infinity

    What it means for a function to approach a number as x grows without bound, the epsilon-M definition, how leading terms decide the behavior of polynomials and rational functions, and where the number e comes from.

    Theory part 19Practice part 20

  2. Infinite Limits

    What it means for a function to grow without bound near a point or far out, how to decide between positive and negative infinity by checking signs, and why infinity is not a real limit value.

    Theory part 21Practice part 22

  3. The Formal Definition of Infinite Limits

    The precise definitions behind limits equal to positive or negative infinity, at a point, from one side and as x grows without bound, and how to prove or refute them.

    Theory part 23Practice part 24

  4. Asymptotes

    How vertical, horizontal and oblique asymptotes are defined through limits, how to find them for rational functions with polynomial division, and why a graph may cross some of them.

    Theory part 25Practice part 26