12.1: Estimating Limits Graphically

1. 12.1: Estimating Limits Graphically Day 1

2. OBJECTIVES Essential Question • Estimate limits of functions at a point. • Estimate limits of functions at infinity. • How can I find the limits of a function numerically • and graphically?

3. Warm-up #1:A Blast from the Past! State the end behavior by filling in the blanks.

4. Warm-up #2:A Blast from the Past! State the end behavior by filling in the blanks.

5. What is a 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 approaches c, is L.

6. Limit f(x) L x c The limit of f(x)… is L. Notation: as x approaches c…

7. Calculating Limits There are three approaches to finding a limit: Numerical Approach – Construct a table of values Graphical Approach – Draw a graph Analytic Approach – Use Algebra or calculus This Lesson Future Lesson

8. Example 1-Numerically Complete the table to find the limit (if it exists). 6.859 7.88 7.988 8 8.012 8.12 9.261 If the function is continuous at the value of x, the limit is easy to calculate.

9. Example 1-Graphically

10. Example 2-Numerically Complete the table to find the limit (if it exists). Can’t divide by 0 -2.1 -2.01 -2.001 DNE -1.999 -1.99 -1.9 If the function is not continuous at the value of x, a graph and table can be very useful.

11. Example 2-Graphically

12. 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.

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

14. Three Limits that Fail to Exist f(x)oscillates between two fixed values as x approaches c. Closest Closer Close

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

17. Example 3-Graphically Given the function t defined by the graph, find the limits at right.

18. Homework: 12.1(day1) Worksheet