Calculus What is “calculus”? What do you learn in a calculus class?

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Presentation transcript:

Calculus What is “calculus”? What do you learn in a calculus class? How do algebra and calculus differ?

10.1 Introduction to Limits Limits form the basis for calculus

One of the basic concepts to the study of calculus is the concept of limit. This concept will help to describe the behavior of f(x) when x is approaching a particular value c. In this section, we will review and learn more about functions, graphs, and limits

When x is closer and closer to 2, F(x) is closer to 3 Therefore: Example 1a: f(x) = 2x - 1 Discuss the behavior of of f(x) when x gets closer to 2 using graph Graph f(x) = 2x - 1 When x is closer and closer to 2, F(x) is closer to 3 Therefore: The limit of f(x) as x approaches 2 is 3 lim(2x-1) = 3 = f(2) X2

Example 1b: f(x) = 2x - 1 Discuss the behavior of the values of f(x) when x gets closer to 2 using table

Exercises Find: lim (x+2) and lim (3x+1) X0 X -1 Do you get 2 and -2? If not, try again

Example 2: Discuss the behavior of f(x) when x gets closer to 2 If x = 2, f(x) is undefined. If you graph, you will see a hole there. x 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5 f (x) 3.5 3.9 3.99 3.999 ? 4.001 4.01 4.1 4.5 Therefore, when x is closer and closer to 2, f(x) is closer to 4 lim f(x) = 4 = f(2) or X2

Example 2: Discuss the behavior of the values of f(x) when x is closer to 2. Does the limit exist? 1 1.9 1.99 2 2.001 2.01 2.1 2.5 f (x) -1 ? * This function is not defined when x = 2. * The limit does not exist because the limit on the left and the limit on the right are not the same. Lim f(x) = -1 represents the limit on the left of 2 Lim f(x) = 1 represents the limit on the right of 2 X2 - X2 +

We write and call K the limit from the left (or left-hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line. and call L the limit from the right (or right-hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line. In order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.