Sandwich TheoremIdea: Suppose we have three functions f, g and h, such that g(x) ? f(x) ? h(x). Then in a graph this would be shown as:Sandwich Theorem:Suppose f, g, and h are functions on an open interval I, containing the value x=c (The functions do not have to be defined at x=c).Suppose further that g(x) ? f(x) ? h(x) for all x?I / {c}and that xc g(x) = xc h(x) = L, then the xc f(x) = L as well.Example:Compute x0 x²cos(1/x)To compute this we use the inequality -1 ? cos(x) ? 1.This is the same as -1 ? cos(1/x) ? 1 (for all x ? 0) and hence, -x² ? x²cos(1/x) ? x² (for all x ? 0).This shows that g(x)= -x², f(x)= x²cos(1/x) and h(x)= x².x0 -x² = 0 and x0 x² = 0 , so the x0 x²cos(1/x) = 0 as well.In this graph the blue line shows x² (h(x)), the green line is -x² (g(x)) and the red line is x²cos(1/x) (f(x)). The graph clearly shows how f(x) is (sandwiched) between the other two graph lines at the point x=0 which enables us to calculate the limit at this point for x²cos(1/x).Example:Compute  x0 sin(x)To compute this we use the inequality -|x| ? sin(x) ? |x|.The graph below shows sin(x) (black line) sandwiched between -|x| (red line) and |x| (green line).x0 -|x| = 0 and x0 |x| = 0, hence the x0 sin(x) = 0 as well. This can be used to solve other examples:Compute x0 cos(x)To compute this we use sin²(x) + cos²(x) = 1,From this we get cos(x) = ?(1-sin²(x))So  x0 cos(x) =  x0 ?(1-sin²(x))  From previous example x0 sin(x) = 0 ,            So  x0 ?1 – (sin²(x)) = ?1-0² = 1So x0 cos(x) = 1.Example:Compute x0 sin(x)/xTo compute this we use the inequality cos(x) ? sin(x)/x ? 1.This shows that g(x)= cos(x), f(x)= sin(x)/x and h(x)= 1.x0 cos(x) = 1 (from previous example) and x0 1 = 1, hence the x0 sin(x)/x = 1 as well.In this graph the green line shows y=1 (h(x)), the red line is cos(x) (g(x)) and the blue line is sin(x)/x (f(x)). The graph clearly shows how f(x) is in between the other two graph lines at the point x=0.Example:If f(x) is a function such that -|2x-4|-2 ? f(x) ? (x-2)²-2, then what is the x2f(x)?To compute this we find the x2 (x-2)²-2 and the x2 -|2x-4|-2:x2 (x-2)²-2 = -2 and the x2 -|2x-4|-2 = -2, so the x2 f(x) = -2 as well.Bibliography:Graphs were created at:Desmos | Graphing Calculator.Available at: https://www.desmos.com/calculator Accessed December 22, 2017.

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