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

 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

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

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

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

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

 -1 ≤ sin θ ≤ 1 for ANY value of θ

 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

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

Evaluate by substitution

 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)

 Test your choices on TI

 Evaluate using substitution and Sandwich Theorem

 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…