One way in which limits are often established is by the Squeeze Theorem.  If and for all x in an open interval that contains (except possibly at ) and = = , then .  We can use this theorem to show that , and to illustrate this fact by graphing. Notice that the "amplitude" of this function is . The curve lies above the graph of , while the curve lies below the graph.

> f:=x->-x^2: g:=x->x^2*cos(20*Pi*x): h:=x->x^2:

> plot([g(x),f(x),h(x)],x=-Pi..Pi,color=[red,green,blue],thickness=[1,3,3]);

> plot([g(x),f(x),h(x)],x=-Pi/4..Pi/4,color=[red,green,blue],thickness=[1,3,3]);

> plot([g(x),f(x),h(x)],x=-Pi/20..Pi/20,color=[red,green,blue],thickness=[1,3,3]);

> plot([g(x),f(x),h(x)],x=-Pi/200..Pi/200,color=[red,green,blue],thickness=[1,3,3]);

To establish the limit using the squeeze theorem, we note that  and , so and . We know that  and , so it follows from the squeeze theorem that

Submission:

(a) Use the squeeze theorem to show that .

(b) Graph your functions , , and on the same set of axes.

Submission worksheet: