Question

(a) By graphing the function

\(f\left( x \right) = \frac{{{\bf{cos2}}x - {\bf{cos}}x}}{{{x^{\bf{2}}}}}\)

and zooming in toward the point where the graph crosses the y-axis, estimate the value of \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{0}}} f\left( x \right)\).

(b) Check your answer in part (a) by evaluating f(x) for values of x that approach 0.

Short answer

a. The graphs are shown below:

b. The value of \(f\left( x \right)\) approaches \( - 1.5\).

Step-by-step solution

Step 1:Find the value of \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{0}}} f\left( x \right)\)

Draw the graph of the function\(f\left( x \right) = \frac{{\cos 2x - \cos x}}{{{x^2}}}\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\left( {\cos 2X - \cos X} \right)/{X^2}\)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.
3. Set the window\( - 6 \le X \le 6\), and\( - 2 \le Y \le 1\).

Visualization of the graph of the function \(f\left( x \right) = \frac{{\cos 2x - \cos x}}{{{x^2}}}\) is shown below:

Now, adjust the window (zoom)\( - 0.5 \le X \le 0.5\), and\( - 2 \le Y \le 1\).

Visualization of the graph of the function \(f\left( x \right) = \frac{{\cos 2x - \cos x}}{{{x^2}}}\) is shown below:

So, the value of the limit is shown below:

\(\begin{array}{c}\mathop {\lim }\limits_{x \to 0} f\left( x \right) &=& \mathop {\lim }\limits_{x \to 0} \frac{{\cos 2x - \cos x}}{{{x^2}}}\\ &=& - \frac{3}{2}\end{array}\)

Thus, \(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = - 1.5\).

Find the values for part (a)

The table below represents the calculation for \(\mathop {\lim }\limits_{x \to 0} f\left( x \right)\).

x	f(x)
\( \pm 0.1\)	\( - 1.493759\)
\( \pm 0.01\)	\( - 1.499938\)
\( \pm 0.001\)	\( - 1.499999\)
\( \pm 0.0001\)	\( - 1.500000\)

So, the value of the limit is \(\mathop {\lim }\limits_{x \to 0} f\left( x \right) = - 1.5\).