1. Definition of Limit, Properties of Limits Section 2.1a

2. Let’s start with an exploration… What are the values of the function given below as x approaches 0??? First, graph the function in the window by Now, look at a table, with TblStart = –0.3 and Tbl = 0.1 –.3.98507 –.2.99335 –.1.99833 0ERROR.1.99833.2.99335.3.99507

3. Let’s start with an exploration… What are the values of the function given below as x approaches 0??? –.3.98507 –.2.99335 –.1.99833 0ERROR.1.99833.2.99335.3.99507 What do these steps suggest? Note: We cannot simply substitute x = 0 into the function, because we’d be dividing by zero…………we need another method…

4. Definition: Limit Let c and L be real numbers. The function f has limit L as x approaches c if, given any positive number, there is a positive number such that for all x, We write which is read, “the limit of f of x as x approaches c equals L.” The notation means that the values of f(x) of the function f approach or equal L as the values of x approach (but do not equal ) the number c…

5. Definition: Limit Let c and L be real numbers. The function f has limit L as x approaches c if, given any positive number, there is a positive number such that for all x, We write As suggested in our opening exploration:

6. Important note: The existence of a limit as x  c never depends on how the function may or may not be defined at c.

7. Definition: Limit Let c and L be real numbers. The function f has limit L as x approaches c if, given any positive number, there is a positive number such that for all x, We write Find each of the following limits:

8. Properties of Limits If L, M, c, and k are real numbers and and, then 1.Sum Rule – The limit of the sum of two functions is the sum of their limits: 2. Difference Rule – The limit of the difference of two functions is the difference of their limits:

9. Properties of Limits If L, M, c, and k are real numbers and and, then 3. Product Rule – The limit of a product of two functions is the product of their limits: 4. Constant Multiple Rule – The limit of a constant times a function is the constant times the limit of the function:

10. Properties of Limits If L, M, c, and k are real numbers and and, then 5. Quotient Rule – The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero:

11. Properties of Limits If L, M, c, and k are real numbers and and, then 6. Power Rule – If r and s are integers,, then provided that is a real number. The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number.

12. Guided Practice Find each of the following limits. (a) (b) Noticeanything?

13. Theorem: Polynomial and Rational Functions 1. Ifis any polynomial function and c is any real number, then 2. Ifare polynomials and c is any real number, then and provided that

14. Guided Practice Find each of the following limits. (a) (b)

15. Guided Practice Explain why you cannot use substitution to determine the given limits. Find the limit if it exists. Cannot use substitution b/c the expression is not defined at x = 0. Since becomes arbitrarily large as x approaches 0 from either side, there is no (finite) limit. Can we support this reasoning graphically???

16. Guided Practice Explain why you cannot use substitution to determine the given limits. Find the limit if it exists. Cannot use substitution b/c the expression is not defined at x = 0. A key rule for evaluating limits: If substitution will not work directly, use algebra techniques to re-write the expression so that substitution will work!!!

17. Guided Practice Explain why you cannot use substitution to determine the given limits. Find the limit if it exists. Cannot use substitution b/c the expression is not defined at x = 0. Support graphically???

18. Guided Practice Determine the given limits algebraically. Support graphically.

19. Guided Practice Determine the given limits algebraically. Support graphically.

20. Guided Practice Determine the given limits algebraically. Support graphically.