2.  Graphically ( √ )  Algebraically (√)  Using the Sandwich Theorem (this lesson)

3.  Used for “ugly” functions when finding the limit using graphical or algebraic methods would be too difficult  Strategy: “sandwich” or “squeeze” the function between two nicer ones and find the limits of the nice functions

4.  If g(x) ≤ f(x) ≤ h(x) and then since f(x) is between g(x) and h(x)

6.  Find two “bread” functions which can “sandwich” f(x)

8.  -1 ≤ cos θ ≤ 1 for ANY value of θ  So g(x) = -x 2 and h(x) = x 2

10.  Division by 0 means we can’t algebraically evaluate… graph is REALLY unpredictable!

11.  -1 ≤ sin θ ≤ 1 for ANY value of θ

12.  We have to change our “bread” slightly… g(x) = -x would cut through f(x) and it is always supposed to stay BELOW…  Fix  use absolute value functions

13.  Once you have your “bread” functions, check on your TI to make sure you have chosen well

14. Evaluate by substitution

17.  Once again, because we don’t want to cut through the function, we are going to use absolute values  (this will happen any time your first choice of “bread” function would cut through the original function)

18.  Test your choices on TI

19.  Evaluate using substitution and Sandwich Theorem

20.  Use the -1 ≤ BLAH ≤ 1 trick and develop your bread functions  If necessary { when your bread has x and not x 2 as the base function } switch to absolute value functions for bread  Use substitution to evaluate bread functions  Therefore, the original limit is…