15.1 Notes

1. 15.1 Notes An introduction to limits and the numerical approach to evaluating limits as x approaches a finite number

2. 15.1 Notes Evaluating a function means to find the value of f(x) for some value of x.

3. 15.1 Notes Evaluating a function means to find the value of f(x) for some value of x. undefined

4. 15.1 Notes A limit problem asks, as x approaches some value, what does f(x) approach? Read “the limit as x approaches 4 of is what?”

5. 15.1 Notes A limit problem asks, as x approaches some value, what does f(x) approach? Read “the limit as x approaches a of f(x) is what?”

6. 15.1 Notes In later lessons, you will learn some methods of evaluating limits. For now, you will use t-tables to help you evaluate limits. You were actually evaluating a limit in today’s “do now” assignment:

7. 15.1 Notes A graphing calculator table feature is useful in evaluating limits. Go to the Y= screen and type in the function.

8. 15.1 Notes Press 2nd and WINDOW (TBL SET) to bring up the “Table Setup Menu.” Start the table at the value at which the limit is being evaluated and set the increment (∆ Tbl) to 0.01 or 0.005.

9. 15.1 Notes Press 2nd and GRAPH (table) to view the table.

10. 15.1 Notes Evaluate the limit:

11. 15.1 Notes Evaluate the limit: 1. 2. 3.

12. 15.1 Notes Because polynomial functions are continuous and have all real numbers as their domain,

13. 15.1 Notes A limit problem does NOT ask what happens when you evaluate a function at some x value. It asks what is happening as x approaches some value. A limit can have an answer as x approaches ‘a’ even when f(a) is not defined. From today’s do now assignment:

14. 15.1 Notes Some limits do not have an answer. That is, f(x) does not approach a value as x approaches some number. We say these limits do not exist. Explanations of the behavior of f(x) in situations where limits do not exist will follow. But first, an explanation of one-sided limits.

15. 15.1 Notes Generally, limits are evaluated by determining the behavior of f(x) as x approaches a number from the left (numbers less than) and from the right (numbers greater than) x.

16. 15.1 Notes To indicate that the behavior of f(x) is going to be determined as x approaches a number from only the left side or only the right side of the number, negative or positive “exponents” are used on the number, respectively. from the left: from the right:

17. 15.1 Notes In order for a limit to exist, the limit as x approaches some value from the left must equal the limit as x approaches that same x value from the right.

18. 15.1 Notes Some limits do not exist because the limit as x approaches some value from the left does not equal the limit as x approaches that same x value from the right.

19. 15.1 Notes

20. 15.1 Notes Each of the above one-sided limits exists. But, since the limit as x approaches 1 from the left does not equal the limit as x approaches 1 from the right, the limit as x approaches 1 does not exist.

21. 15.1 Notes Some limits do not exist because as x approaches some number f(x) increases without bound (goes to infinity) or decreases without bound (goes to negative infinity).

22. 15.1 Notes DNE Notice the use of arrows rather than equal signs.

23. 15.1 Notes Some limits do not exist because f(x) oscillates between small and large and/or positive and negative values as x approaches some number.

24. 15.1 Notes

25. 15.1 Notes Evaluate the limit. If the limit does not exist, explain why not. 1. 2. 3.