Question

Evaluate the function

\(f\left( x \right) = {x^2} - \left( {{{{2^x}} \mathord{\left/{\vphantom{{{2^x}} {1000}}}\right.}{1000}}}\right)\)

forx=1,0.8,0.6,0.4,0.2,0.1, and0.05, and guess the value of \(\mathop {\lim }\limits_{x \to 0} \left( {{x^2} - \frac{{{2^x}}}{{1000}}} \right)\)

(b) Evaluate f(x) for x=0.04,\0.02,\0.01,\0.005,\0.003,and0.001.Guess again.

Short answer

(a) The values of \(f\left( x \right)\) for the given values of \(x\) are \(0.998000\), \(0.638259\), \(0.358484\), \(0.158680\), \(0.038851\), \(0.008928\), and \(0.001465\), and \(\mathop {\lim }\limits_{x \to 0} \left( {{x^2} - \frac{{{2^x}}}{{1000}}} \right) = 0\).

(b) The values of \(f\left( x \right)\) for the given values of \(x\) are\(0.000572\), \( - 0.000614\), \( - 0.000907\), \( - 0.000978\), \( - 0.000993\), and \( - 0.001000\), and \(\mathop {\lim }\limits_{x \to 0} \left( {{x^2} - \frac{{{2^x}}}{{1000}}} \right) = - 0.001\)

Step-by-step solution

Describe the given information

The given function is as follows:

\(f\left(x\right)={x^2}-\left({{{{2^x}}\mathord{\left/{\vphantom{{{2^x}}{1000}}}\right.-}{1000}}}\right)\)

Find the value of the function for the given values of x, and also guess the value of the given limit

Substitute \(x = 1\) in function\(f\left( x \right) = {x^2} - \frac{{{2^x}}}{{1000}}\).

\(\begin{aligned}f\left( x \right) &= {\left( 1 \right)^2} - \frac{{{2^1}}}{{1000}}\\ &= 1 - \frac{2}{{1000}}\\ &= 1 - 0.002\\ &= 0.998\end{aligned}\)

Similarly, calculate the other values of \(f\left( x \right)\) for the different values of \(x\) as shown in the following table.

\(x\)	\(f\left( x \right) = {x^2} - \frac{{{2^x}}}{{1000}}\)
\(1\)	\(0.998000\)
\(0.8\)	\(0.638259\)
\(0.6\)	\(0.358484\)
\(0.4\)	\(0.158680\)
\(0.2\)	\(0.038851\)
\(0.1\)	\(0.008928\)
\(0.05\)	\(0.001465\)

Using the table guess the value of\(\mathop {\lim }\limits_{x \to 0} \left( {{x^2} - \frac{{{2^x}}}{{1000}}} \right)\).

\(\mathop {\lim }\limits_{x \to 0} \left( {{x^2} - \frac{{{2^x}}}{{1000}}} \right) = 0\)

Therefore, the values of \(f\left( x \right)\) for the given values of \(x\) are \(0.998000\), \(0.638259\), \(0.358484\), \(0.158680\), \(0.038851\), \(0.008928\), and \(0.001465\), and \(\mathop {\lim }\limits_{x \to 0} \left( {{x^2} - \frac{{{2^x}}}{{1000}}} \right) = 0\).

Find the value of the function for the given values of x, and also guess the value of the given limit

Substitute \(x = 0.04\) in fun

ction\(f\left( x \right) = {x^2} - \frac{{{2^x}}}{{1000}}\).

\(\begin{aligned}f\left( x \right) &= {\left( {0.04} \right)^2} - \frac{{{2^{0.04}}}}{{1000}}\\ &= 0.0016 - \frac{{{2^{0.04}}}}{{1000}}\\ &= 0.000572\end{aligned}\)

Similarly, calculate the other values of \(f\left( x \right)\) for the different values of \(x\) as shown in the following table.

\(x\)	\(f\left( x \right) = {x^2} - \frac{{{2^x}}}{{1000}}\)
\(0.04\)	\(0.000572\)
\(0.02\)	\( - 0.000614\)
\(0.01\)	\( - 0.000907\)
\(0.005\)	\( - 0.000978\)
\(0.003\)	\( - 0.000993\)
\(0.001\)	\( - 0.001000\)

Guess the value of\(\mathop {\lim }\limits_{x \to 0} \left( {{x^2} - \frac{{{2^x}}}{{1000}}} \right)\).

From the table, it can be observed that as \(x\)tends to 0 then the value of the function approaches to\( - 0.001\).

Therefore, the values of \(f\left( x \right)\) for the given values of \(x\) are \(0.000572\), \( - 0.000614\), \( - 0.000907\), \( - 0.000978\), \( - 0.000993\), and \( - 0.001000\), and \(\mathop {\lim }\limits_{x \to 0} \left( {{x^2} - \frac{{{2^x}}}{{1000}}} \right) = - 0.001\).