1. Section 1.2 - Finding Limits Graphically and Numerically

2. Limit Informal Definition: If f(x) becomes arbitrarily close to a single REAL number L as x approaches c from either side, the limit of f(x), as x appraches c, is L. The limit of f(x)… as x approaches c… is L. Notation: c L f(x)f(x) x

3. Calculating Limits Our book focuses on three ways: 1.Numerical Approach – Construct a table of values 2.Graphical Approach – Draw a graph 3.Analytic Approach – Use Algebra or calculus This Lesson Next Lesson

4. Example 1 Use the graph and complete the table to find the limit (if it exists). x 1.91.991.99922.0012.012.1 f(x)f(x) 6.8597.88 7.98889.2618.128.012 If the function is continuous at the value of x, the limit is easy to calculate.

5. Example 2 Use the graph and complete the table to find the limit (if it exists). x -1.1-1.01 -1.001 -.999-.99-.9 f(x)f(x) -2.1-2.01 -2.001DNE-1.9-1.99-1.999 If the function is not continuous at the value of x, a graph and table can be very useful. Can’t divide by 0

6. -6 Example 3 Use the graph and complete the table to find the limit (if it exists). x -4.1-4.01 -4.001 -4 -3.999 -3.99-3.9 f(x)f(x) 2.92.99 2.99982.92.992.999 If the function is not continuous at the value of x, the important thing is what the output gets closer to as x approaches the value. The limit does not change if the value at -4 changes. -6

7. Three Limits that Fail to Exist f(x) approaches a different number from the right side of c than it approaches from the left side.

8. Three Limits that Fail to Exist f(x) increases or decreases without bound as x approaches c.

9. Three Limits that Fail to Exist f(x) oscillates between two fixed values as x approaches c. x 0 f(x)f(x) 1 DNE 11 Close Closer Closest

10. A Limit that DOES Exist If the domain is restricted (not infinite), the limit of f(x) exists as x approaches an endpoint of the domain.

11. Example 1 Given the function t defined by the graph, find the limits at right.

12. Example 2 Sketch a graph of the function with the following characteristics: 1. does not exist, Domain: [-2,3), and Range: (1,5) 2. does not exist, Domain: (-∞,-4)U(-4,∞), and Range: (-∞,∞)

13. Classwork Sketch a graph and complete the table to find the limit (if it exists). x -0.1-0.01 -0.001 00.0010.010.1 f(x)f(x) 2.86802.732 2.7196DNE2.59372.70482.7169 This a very important value that we will investigate more in Chapter 5. It deals with natural logs. Why is there a lot of “noise” over here?