Question

Use the Squeeze theorem to show that

\(\mathop {\lim }\limits_{x \to 0} {x^2}\cos 20\pi x = 0\)

Illustrate by graphing the function \(f\left( x \right) = - {x^2}\), \(g\left( x \right) = {x^2}\cos 20\pi x\), and \(h\left( x \right) = {x^2}\) on the same screen.

Short answer

It is proved that \(\mathop {\lim }\limits_{x \to 0} {x^2}\cos 20\pi x = 0\). The graph of the functions is obtained

Step-by-step solution

The Squeeze theorem

When\(f\left( x \right) \le g\left( x \right) \le h\left( x \right)\), if\(x\)is near\(a\)(except possibly at\(a\)) and

\(\mathop {\lim }\limits_{x \to a} f\left( x \right) = \mathop {\lim }\limits_{x \to a} h\left( x \right) = L\) then \(\mathop {\lim }\limits_{x \to a} g\left( x \right) = L\)

Show that \(\mathop {\lim }\limits_{x \to 0} {x^2}\cos 20\pi x = 0\)

Suppose that \(f\left( x \right) = - {x^2}\), \(g\left( x \right) = {x^2}\cos 20\pi x\) and \(h\left( x \right) = {x^2}\). Then;

\( - 1 \le \cos 20\pi x \le 1\)

Multiply the above inequality by \({x^2}\) as shown below:

\(\begin{array}{c} - {x^2} \le {x^2}\cos 20\pi x \le {x^2}\\f\left( x \right) \le g\left( x \right) \le h\left( x \right)\end{array}\)

Therefore, according to the Squeeze theorem, that\(\mathop {\lim }\limits_{x \to 0} g\left( x \right) = 0\)because\(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = \mathop {\lim }\limits_{x \to 0} h\left( x \right)\).

Thus, it is proved that \(\mathop {\lim }\limits_{x \to 0} {x^2}\cos 20\pi x = 0\).

Illustrate by graphing the functions on the same screen

The procedure to draw the graph of the function \(f\left( x \right) = - {x^2},\)\(h\left( x \right) = {x^2}\cos 20\pi x\), and \(h\left( x \right) = {x^2}\) by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\( - {X^2}\)in the\({Y_1}\)tab.
2. Enter the equation\({X^2}\cos 20\pi X\)in the\({Y_2}\)tab.
3. Enter the equation\({X^2}\)in the\({Y_3}\)tab.
4. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the functions as shown below:

Thus, the graph of the functions is obtained.