Question

ing the values of the function for the inputs listed in this table.
x | f(x)
0.2 |
0.1 |
0.05 |
0.01 |
0.001 |
0.0001 |
0.00001 |
values in this table, it appears \\(\lim\limits_{x\to 0} \frac{\cos(7x) - \cos(4x)}{x^2} = \\)

Explanation:

Step1: Define the function

$f(x) = \frac{\cos(7x) - \cos(4x)}{x^2}$

Step2: Calculate $f(0.2)$

Substitute $x=0.2$:
$f(0.2) = \frac{\cos(7\times0.2) - \cos(4\times0.2)}{(0.2)^2} = \frac{\cos(1.4) - \cos(0.8)}{0.04} \approx \frac{0.169968 - 0.696707}{0.04} \approx -13.1685$

Step3: Calculate $f(0.1)$

Substitute $x=0.1$:
$f(0.1) = \frac{\cos(7\times0.1) - \cos(4\times0.1)}{(0.1)^2} = \frac{\cos(0.7) - \cos(0.4)}{0.01} \approx \frac{0.764842 - 0.921061}{0.01} \approx -15.6219$

Step4: Calculate $f(0.05)$

Substitute $x=0.05$:
$f(0.05) = \frac{\cos(7\times0.05) - \cos(4\times0.05)}{(0.05)^2} = \frac{\cos(0.35) - \cos(0.2)}{0.0025} \approx \frac{0.939372 - 0.980067}{0.0025} \approx -16.278$

Step5: Calculate $f(0.01)$

Substitute $x=0.01$:
$f(0.01) = \frac{\cos(7\times0.01) - \cos(4\times0.01)}{(0.01)^2} = \frac{\cos(0.07) - \cos(0.04)}{0.0001} \approx \frac{0.997551 - 0.999200}{0.0001} \approx -16.49$

Step6: Calculate $f(0.001)$

Substitute $x=0.001$:
$f(0.001) = \frac{\cos(7\times0.001) - \cos(4\times0.001)}{(0.001)^2} = \frac{\cos(0.007) - \cos(0.004)}{0.000001} \approx \frac{0.9999755 - 0.999992}{0.000001} \approx -16.5$

Step7: Calculate $f(0.0001)$

Substitute $x=0.0001$:
$f(0.0001) = \frac{\cos(7\times0.0001) - \cos(4\times0.0001)}{(0.0001)^2} = \frac{\cos(0.0007) - \cos(0.0004)}{0.00000001} \approx \frac{0.999999755 - 0.99999992}{0.00000001} \approx -16.5$

Step8: Calculate $f(0.00001)$

Substitute $x=0.00001$:
$f(0.00001) = \frac{\cos(7\times0.00001) - \cos(4\times0.00001)}{(0.00001)^2} = \frac{\cos(0.00007) - \cos(0.00004)}{0.0000000001} \approx \frac{0.99999999755 - 0.9999999992}{0.0000000001} \approx -16.5$

Step9: Identify the limit

As $x\to0$, $f(x)$ approaches $-16.5 = -\frac{33}{2}$

Answer:

Table of values:

$x$	$f(x)$
0.1	$-15.6219$
0.05	$-16.278$
0.01	$-16.49$
0.001	$-16.5$
0.0001	$-16.5$
0.00001	$-16.5$

Limit:

$\lim_{x\to0} \frac{\cos(7x) - \cos(4x)}{x^2} = -\frac{33}{2}$ (or $-16.5$)