Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 37

(a) Evaluate the function $f(x) = x^2 - (2^x/1000)$ for $x = 1 $, 0.8, 0.6, 0.4, 0.2, 0.1 and 0.05. and guess the value of $\displaystyle \lim \limits_{x \to 0} \left( x^2 - \frac{2^x}{1000} \right)$


$
\begin{array}{|c|c|}
\hline
x & f(x)\\
\hline
1 & 0.998\\
0.8 & 0.638\\
0.6 & 0.358\\
0.4 & 0.159\\
0.2 & 0.039\\
0.1 & 0.009\\
0.05 & 0.001\\
\hline
\end{array}
$


Based from the vlaues we obtain in the table; as $x$ approaches 0, the value of $f(x)$ approaches 0 as well. Therefore $\displaystyle \lim \limits_{x \to 0} \left( x^2 - \frac{2^x}{1000} \right) = 0$



(b) Evaluate $f(x)$ for $x = $ 0.04, 0.02, 0.01, 0.005, 0.003 and 0.001. Guess again


$
\begin{array}{|c|c|}
\hline
x & f(x)\\
\hline
0.04 & 0.00057\\
0.02 & -0.00061\\
0.01 & -0.00091\\
0.005 & -0.00098\\
0.003 & -0.00099\\
0.001 & -0.001\\
\hline
\end{array}
$



Based from the values we obtain from the table; The $\lim\limits_{x \to 0} f(x)$ seems to have a value of -0.001.