1.0 Introduction To Limits (had)

1.0 Introduction to Limits (Had)

• 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

Example 1a: f(x) = 2x - 1 Discuss the behavior of of f(x) when x is close 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 is close to 2 using table

x

F(x)

1.5

1.9

1.99

1.999

2

2.001 2.01

2.1

2.5

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

Try these : • Find: lim (x+2) and lim (3x+1) using graph X0

X -1

What is your conclusion in finding limit of a function? DO SUBSTITUTION!!!

x2  4 f ( x)  x2

Note : Special Situation #1

Example 2: Discuss the behavior of f(x) when x is closer to 2

x f (x)

1.5

1.9

1.99

1.999

2

2.001

2.01

2.1

2.5

x2  4 f ( x)  x2

Note : Special Situation #1

Example 2: Discuss the behavior of f(x) when x is 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)

X2

or

Special Situation #1 : if by substituting, you get 0/0 then something has to be done to the function before substitution. In this case what should you do?

Situation #2

f ( x) 

x2 x2

Example 2: Discuss the behavior of the values of f(x) when x is closer to 2. Does the limit exist? x

0

1

1.9

1.99

2

f (x)

-1

-1

-1

-1

?

2.001 2.01 2.1

* 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 X2 -

Lim f(x) = 1 represents the limit on the right of 2 X2 +

1

1

1

2.5 1

■ 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. notasi

■ We write 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. notasi

■ In order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.