Understanding Limits of Functions: Theory and Application

The limit of f(x) as x approaches c is ______

The limit, denoted by L, is the unique real value that f(x) will approach as x approaches ______

To evaluate the given limit lim (1 + 3x) as x approaches 2, we will make use of a ______ to help us keep track of the effect

On the number line, x may approach 2 through values on its ______

When x approaches 2 from its left, the values chosen should be close to ______

The limit will be able to describe clearly what is happening to the function near that ______

Distinguishing between lim f (x) and f (c) using a table of values x!c 4. Distinguishing between lim f (x) and f (c) using the graph of y = f (x) x!c 5. Enumeration of the eight basic limit theorems 6. Application of the eight basic limit theorems on simple examples 7. Limits of polynomial functions 8. Limits of rational functions 9. Limits of radical functions 10. Intuitive notions of infinite limits 2 TOPIC 1.1: The Limit of a Function DEVELOPMENT OF THE LESSON (A) ACTIVITY In order to find out what the students’ idea of a limit is, ask them to bring cutouts of news items, articles, or drawings which for them illustrate the idea of a limit. These may be posted on a wall so that they may see each other’s homework, and then have each one explain briefly why they think their particular cutout represents a limit. (B) INTRODUCTION Limits are the backbone of calculus, and calculus is called the Mathematics of Change. The study of limits is necessary in studying change in great detail. The evaluation of a particular limit is what underlies the formulation of the derivative and the integral of a function. For starters, imagine that you are going to watch a basketball game. When you choose seats, you would want to be as close to the action as possible. You would want to be as close to the players as possible and have the best view of the game, as if you were in the basketball court yourself.

Sketch one possible ______ of a function f(x) defined on R that satisfies all the listed conditions.

Critical to the study of limits is the understanding that the value of lim f(x) as x approaches c may be distinct from the value of the function at x = c, that is, f(c). As seen in previous examples, the limit may be evaluated at values not included in the domain of f. Thus, it must be clear to a student of calculus that the exclusion of a value from the domain of a function does not prohibit the evaluation of the limit of that function at that excluded value, provided of course that f is defined at the points near c. In fact, these cases are actually the more interesting ones to investigate and evaluate. Furthermore, the awareness of this distinction will help the student understand the concept of ______, which will be tackled in Lessons 3 and 4.

We will mostly recall our discussions and examples in Lesson 1. Let us again consider lim (1 + 3x). Write 6 'fill in the blank' statements using the content above. Provide the missing word as the answer. Focus on topics: limits, trigonometric functions, exponential functions, ______, evaluating limits.

Possible answer (there are many other possibilities): y 6 5 4 3 2 1 2 1 0 1 2 16 3 4 5 6 x TOPIC 1.2: The Limit of a Function at c versus the Value of the Function at c DEVELOPMENT OF THE LESSON (A) INTRODUCTION Critical to the study of limits is the understanding that the value of lim f(x) x!c may be distinct from the value of the function at x = c, that is, f(c). As seen in previous examples, the limit may be evaluated at values not included in the domain of f. Thus, it must be clear to a student of calculus that the exclusion of a value from the domain of a function does not prohibit the evaluation of the limit of that function at that excluded value, provided of course that f is defined at the points near c. In fact, these cases are actually the more interesting ones to investigate and evaluate. Furthermore, the awareness of this distinction will help the student understand the concept of ______, which will be tackled in Lessons 3 and 4. (B) LESSON PROPER We will mostly recall our discussions and examples in Lesson 1. Let us again consider lim (1 + 3x).

Lim f(x) = 1 as x approaches ______

F(4) = ______