Topic: Limits and Continuity

Subtopic: Limits Graphically

**Overview**

The first big topic in calculus is limits. As x gets closer and closer to a number, f(x) gets closer and closer (approaches) a value. This is the concept of a limit. The answer from the limit is the value that f(x) is approaching.

In this lesson we analyze limits graphically. We'll observe the graph of f(x) and see what value the outputs are approaching as the input x gets closer and closer to a given number in an intuitive manner.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 1.1.1 Write limits using correct mathematical notation
- 1.1.2 Graphically determine the limit of a function when the limit is a real number
- 1.1.3 Graphically determine the limit of a function when the limit is a infinite
- 1.1.4 Graphically determine that a limit of a function does not exist
- 1.1.5 Understand the difference between a function being undefined at an x-value vs. the limit not existing at an x-value (e.g. holes vs. jumps)
- 1.1.6 Graphically determine a one-sided limit (limit from the left or a limit from the right)

**Terminology**

Define: limit, approach (in terms of a limit), D.N.E. = does not exist, one-sided limit

**Text Notes**

- Chapter 1 is all review from a precalculus course. Skim and review if needed. If this material is not familiar you should retake College Algebra (Clark’s math 111) and/or College Trigonometry (Clark’s math 103) before taking this Calculus I course. Being an early transcendentals course, you must be confident with functions and graphs including exponential, logarithmic, and trigonometric starting on day one.
- Chapter 2 begins with an overview of calculus and its applications. It serves as an important foundation for the entire course, but is a bit overwhelming for our first section of material. Many of the concepts are covered in more depth as we progress through chapter 2.

**Supplemental Resources (recommended)**

Watch What is Calculus?, Michael Harrison (Socratica) and What is Calculus Used For?, Jeff Heys, TEDx Talk.

**Supplemental Resources (optional)**

Watch What is Calculus? by James Sousa (MathIsPower4U) or read Why Do We Study Calculus? by Prof. Eric Schechter of Vanderbilt University.

Video: Calculating Limits Intuitively, Selwyn Hollis's Video Calculus

Lesson: Slopes and Velocities, Dale Hoffman's Contemporary Calculus overviews several applications of calculus for motivational purposes. You are NOT expected to know solve these problems ... yet!

Video Examples: MathIsPower4U | Calculus I is an excellent resource of short tutorial video examples particularly in instances when you need an additional example of a specific type of problem. This week watching any of the videos listed under "Limits" category may be helpful.