1. Finding Limits Algebraically Chapter 2: Limits and Continuity

2. What you’ll learn about Finding a limit algebraically (or analytically) Using properties of limits Finding a limit using the sandwich (or squeeze) theorem Two special limits Limits of piecewise functions

3. Direct Substitution

4. Properties of Limits

5. Product Rule: Constant Multiple Rule: Properties of Limits continued

7. Polynomial and Rational Functions Trying to evaluate functions using just these rules can be a grueling process (I’m not going to make you do that) so it is a little easier to just combine a bunch of those rules to evaluate polynomial and rational functions. Now let’s look at a few examples using these new properties and rules.

8. Example Limits Use the rules from the previous slides to find the following limits using direct substitution.

9. Evaluating Limits As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. This includes trigonometric, exponential, and logarithmic functions, and composites of these functions. Now lets try an example: Let’s look at the graph first: If we can, we always want to confirm this algebraically:

10. Evaluating Limits cont. Now let’s try another example:

11. Dividing Out Technique Sometimes direct substitution fails even when the limit exists. This is one technique to help find the limit in that case. Let’s use it in an example One approach we can try to do is factor our original function and see if we can cancel anything out that may allow us to do direct substitution.

12. Rationalizing Technique Another technique that we can try when direct substitution fails but a limit appears to exist is to rationalize the numerator or denominator and see if that helps us to make a cancellation that allows for us to use direct substitution. Now we can easily find the limit using direct substitution.

13. Recap – Finding Limits Algebraically

14. A General Strategy for Finding Limits 1)Try direct substitution. Direct substitution will work for most types of functions. For piecewise functions, if you are looking for the limit at an x value where the rule for the function changes, use direct substitution on the left and right rules at that point and see if they are equal. Remember, in order for a limit to exist, both of the one-sided limits must be equal to each other. 2)If you can’t evaluate the limit of f(x) at a point c using direct substitution, try dividing out or rationalizing to find a new function g that agrees with f for all x other than x = c. Choose a g where you can find the limit at x = c using direct substitution. 3)Always confirm or reinforce your conclusion using a graph or table of values or both if you can. It doesn’t matter if you check before or after you try direct substitution, but it’s very important to make sure that your answer makes sense. If you don’t have calculator access, you just have to trust your algebra.

15. The Sandwich (or Squeeze) Theorem

16. Sandwich Theorem Example

17. Two Special Trigonometric Limits

18. Example With a Piecewise Function Remember that we can only say that a limit exists at a given x value if both of the one-sided limits exist and are equal at that x value. In this case, we will use direct substitution in both of the given definitions of f(x) since x=2 is the value where the definition of f(x) changes. The symbols representation in the box above is what is expected of you on my exams and on the AP exam for Free-Response questions as justification.

19. Summary