Aim: What is the limit?

1. Aim: What is the limit? Do Now: Evaluate the functions 1. f(x) = 3x – 2, when x = 2 2. , when x = 1 3. when x = 0

2. 1. f(2) = 6 – 2 = 4 2. f(1) is undefined We can not evaluate f(1) by direct substitution, but we can estimate f(x) reaches 2 when x approaches 1 3. f(x) getting larger and larger when x approaches to 0 Definition of Limit If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L. This is written as

3. The limit of f(x) as x approaches 2 can be obtained by direct substitution. The limit of f(x) as x approaches 1 can not be obtained by direct substitution, but we can estimate or by factoring. In this case, the limit of f(x) does not exist, since f(x) reaches infinity as x approaches 0

4. We can conclude that the limit of a function f(x) as x approaches c sometimes exists and sometimes not, depending on the function and the value of c There are different methods to evaluate the limit of a function if the limit exist. • Direct substitution • Cancellation • Rationalize The following examples will illustrate those three situations

5. Direct substitution Cancellation Rationalization By direct substitution we get

6. In this case, we can rewrite the function by rationalizing the numerator

7. When does a limit does not exist? 1. When the left and right limit are not equal y . 1 x y 2. Unbounded behavior x

8. 3. Oscillating behavior y x

9. Evaluate the limit 1. 2. 4 1 3. 4. If f(x) = x2 – 3x find 2x – 3