1. 16.2 Limit Theorems

2. There are theorems that allow us to calculate limits quickly. Limit Theorems (Part 1) If f (x) is equal to a constant k, then (limit of constant is constant) If f (x) = x m, where m is a positive real number, then (limit of x to power is # to power) Ex 1) Evaluate each limit (Two Truths & a Lie) should be –27

3. Knowing these 2 basic rules, we can combine them to get more intricate functions utilizing these next rules. Limit Theorems (Part 2) (limit of a sum is sum of limits) (limit of a diff is diff of limits) (limit of product is product of limits) (limit of a quotient is quotient of limits)

4. These theorems make it easier to evaluate Instead of this: We can go straight to substituting Ex 2) Find So much writing!

5. So, what happens if we want the limit at a value that makes the function undefined? The limit might not exist or you may be able to work some algebraic magic! Ex 3) Find the limit Can’t factor Think about graph  Asymptote at x = –2 So limit does not exist Can factor! *remember, it is actually undefined here, but it approaches 4*

6. Recall that This helps us evaluate limits that go to positive or negative infinity. Ex 4) Find the limit.

7. *Note: horizontal asymptote rules & limits compliment each other if degree of f (x) = degree of g(x)  limit is ratio of leading coefficients aka horizontal asymptote if degree of f (x) < degree of g(x)  limit is 0 aka horizontal asymptote if degree of f (x) > degree of g(x)  limit is + or –∞ aka no asymptote Recall a continuous function – don’t lift up your pencil, no holes, etc. Now, a more formal definition. A function f (x) is continuous at x = c iff to be continuous all 3 must be met! Thm: Polynomial functions are continuous at every real number c.

8. Ex 5) Determine if the function is continuous for all real values of x. If not, indicate which condition of the definition failed. (Draw the graph!) f (–1) is definedf (–1) = 1 But the limit does not exist not continuous

9. Homework #1602 Pg 863 #1, 3, 4, 5, 7, 9, 11, 13, 15, 18–21, 23, 25, 27, 29, 32–34, 36–38